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Geometry Section 2-2

J
Jimbo Lamb

This document defines key vocabulary terms related to statements, conditionals, and biconditionals. It defines statements, truth values, negation, and compound statements. It then defines conjunctions, disjunctions, if-then statements, hypotheses, conclusions, related conditionals, converses, inverses, contrapositives, and biconditionals. It provides examples of writing compound statements and determining their truth values. It also provides examples of identifying hypotheses and conclusions of conditional statements and writing them in if-then form. The document aims to build foundational understanding of propositional logic.

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Section 2-2 Statements, Conditionals, and Biconditionals
Essential Questions • How do you write compound statements and determine truth values of compound statements? • How do you write conditional statements and determine truth values of conditional statements?
Vocabulary 1. Statement: 2. Truth Value: 3. Negation: 4. Compound Statement:
Vocabulary 1. Statement: A sentence that is either true or false, often represented as statements p and q 2. Truth Value: 3. Negation: 4. Compound Statement:
Vocabulary 1. Statement: A sentence that is either true or false, often represented as statements p and q 2. Truth Value: The statement is either true (T) or false (F) 3. Negation: 4. Compound Statement:
Vocabulary 1. Statement: A sentence that is either true or false, often represented as statements p and q 2. Truth Value: The statement is either true (T) or false (F) 3. Negation: A statement with the opposite meaning and opposite truth value; the negation of p is ~p 4. Compound Statement:
Vocabulary 1. Statement: A sentence that is either true or false, often represented as statements p and q 2. Truth Value: The statement is either true (T) or false (F) 3. Negation: A statement with the opposite meaning and opposite truth value; the negation of p is ~p 4. Compound Statement: Two or more statements joined by and or or
Vocabulary 5. Conjunction: 6. Disjunction: 7. Conjunction: 8. If-Then Statements:
Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: 7. Conjunction: 8. If-Then Statements:
Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: A compound statement using the word or or the symbol ⋁ 7. Conjunction: 8. If-Then Statements:
Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: A compound statement using the word or or the symbol ⋁ 7. Conjunction: A statement that fits the if-then form, providing a connection between the two phrases 8. If-Then Statements:
Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: A compound statement using the word or or the symbol ⋁ 7. Conjunction: A statement that fits the if-then form, providing a connection between the two phrases 8. If-Then Statements: Another name for a conditional statement; in the form of if p, then q
Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: A compound statement using the word or or the symbol ⋁ 7. Conjunction: A statement that fits the if-then form, providing a connection between the two phrases 8. If-Then Statements: Another name for a conditional statement; in the form of if p, then q p → q
Vocabulary 9. Hypothesis: 10: Conclusion: 11. Related Conditionals: 12. Converse:
Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: 11. Related Conditionals: 12. Converse:
Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: The phrase that is the then part of the conditional; q 11. Related Conditionals: 12. Converse:
Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: The phrase that is the then part of the conditional; q 11. Related Conditionals: Statements that are based off of a given conditional statement 12. Converse:
Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: The phrase that is the then part of the conditional; q 11. Related Conditionals: Statements that are based off of a given conditional statement 12. Converse: A statement that is created by switching the hypothesis and conclusion of a conditional
Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: The phrase that is the then part of the conditional; q 11. Related Conditionals: Statements that are based off of a given conditional statement 12. Converse: A statement that is created by switching the hypothesis and conclusion of a conditional q → p
Vocabulary 13. Inverse: 14. Contrapositive: 15. Logically Equivalent:
Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: 15. Logically Equivalent:
Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: ~ p →~ q 15. Logically Equivalent:
Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q 15. Logically Equivalent:
Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q ~ q →~ p 15. Logically Equivalent:
Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q ~ q →~ p 15. Logically Equivalent: Statements with the same truth values
Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q ~ q →~ p 15. Logically Equivalent: Statements with the same truth values A conditional and its contrapositive
Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q ~ q →~ p 15. Logically Equivalent: Statements with the same truth values A conditional and its contrapositive The converse and inverse of a conditional
Vocabulary 16. Biconditional:
Vocabulary 16. Biconditional: A true compound statement that consist of the conditional and its converse
Vocabulary 16. Biconditional: A true compound statement that consist of the conditional and its converse p ↔ q
Vocabulary 16. Biconditional: A true compound statement that consist of the conditional and its converse p ↔ q “p if and only if q,” or “p IFF q”
Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points a. p and q
Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points a. p and q One meter is 100 mm and November has 30 days
Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points a. p and q One meter is 100 mm and November has 30 days q is true, but p is false, so the conjunction is false
Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points b. ~p ⋀ r
Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points b. ~p ⋀ r One meter is not 100 mm and a line is defined by two points
Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points b. ~p ⋀ r One meter is not 100 mm and a line is defined by two points Both ~p and r are true, so ~p ⋀ r is true
Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number a. p or q AB ! "!!
Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number a. p or q AB ! "!! is proper notation for “ray AB” or kilometers are metric units AB ! "!!
Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number a. p or q AB ! "!! Both p and q are true, so p or q is true is proper notation for “ray AB” or kilometers are metric units AB ! "!!
Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number b. q ⋁ r AB ! "!!
Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number b. q ⋁ r AB ! "!! Kilometers are metric units or 15 is a prime number
Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number b. q ⋁ r AB ! "!! Since one of the statements (q) is true, q ⋁ r is true Kilometers are metric units or 15 is a prime number
Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number c. ~p ⋁ r AB ! "!!
Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number c. ~p ⋁ r AB ! "!! is not proper notation for “ray AB” or 15 is a prime number AB ! "!!
Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number c. ~p ⋁ r AB ! "!! Since both ~p and r are false, ~p ⋁ r is false is not proper notation for “ray AB” or 15 is a prime number AB ! "!!
Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis Conclusion b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis Conclusion b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game. Hypothesis
Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis Conclusion b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game. Hypothesis Conclusion
Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a. Measured distance is positive.
Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a. Measured distance is positive. Hypothesis: A distance is measured
Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a. Measured distance is positive. Hypothesis: A distance is measured Conclusion: It is positive
Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a. Measured distance is positive. Hypothesis: A distance is measured Conclusion: It is positive If a distance is measured, then it is positive.
Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. b. A six-sided polygon is a hexagon
Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. b. A six-sided polygon is a hexagon Hypothesis: A polygon has six sides
Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. b. A six-sided polygon is a hexagon Hypothesis: A polygon has six sides Conclusion: It is a hexagon
Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. b. A six-sided polygon is a hexagon Hypothesis: A polygon has six sides Conclusion: It is a hexagon If a polygon has six sides, then it is a hexagon.
Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. a. If you subtract a whole number from another whole number, the result is also a whole number.
Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. a. If you subtract a whole number from another whole number, the result is also a whole number. False
Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. a. If you subtract a whole number from another whole number, the result is also a whole number. False 5 − 11 = −6
Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. b. If last month was September, then this month is October.
Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. b. If last month was September, then this month is October. True
Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. c. When a rectangle has an obtuse angle, it is a parallelogram.
Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. c. When a rectangle has an obtuse angle, it is a parallelogram. A rectangle cannot have an obtuse angle, so we cannot test this. All rectangles are parallelograms.
Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. c. When a rectangle has an obtuse angle, it is a parallelogram. True: False Hypothesis A rectangle cannot have an obtuse angle, so we cannot test this. All rectangles are parallelograms.
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse:
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse: If MN ≅ NO, then N is the midpoint of MO.
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse: If MN ≅ NO, then N is the midpoint of MO. False
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse: If MN ≅ NO, then N is the midpoint of MO. False M, N, and O might not be collinear
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse: If MN ≅ NO, then N is the midpoint of MO. False M, N, and O might not be collinear M N O
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse:
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse: If N is not the midpoint of MO, then MN ≅ NO.
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse: False If N is not the midpoint of MO, then MN ≅ NO.
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse: False If N is not the midpoint of MO, then MN ≅ NO. If N is not on MO, then MN could be congruent to NO.
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse: False If N is not the midpoint of MO, then MN ≅ NO. If N is not on MO, then MN could be congruent to NO. M N O
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Contrapositive:
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Contrapositive: If MN ≅ NO, then N is not the midpoint of MO.
Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Contrapositive: True If MN ≅ NO, then N is not the midpoint of MO.

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Geometry Section 2-2

  • 2. Essential Questions • How do you write compound statements and determine truth values of compound statements? • How do you write conditional statements and determine truth values of conditional statements?
  • 3. Vocabulary 1. Statement: 2. Truth Value: 3. Negation: 4. Compound Statement:
  • 4. Vocabulary 1. Statement: A sentence that is either true or false, often represented as statements p and q 2. Truth Value: 3. Negation: 4. Compound Statement:
  • 5. Vocabulary 1. Statement: A sentence that is either true or false, often represented as statements p and q 2. Truth Value: The statement is either true (T) or false (F) 3. Negation: 4. Compound Statement:
  • 6. Vocabulary 1. Statement: A sentence that is either true or false, often represented as statements p and q 2. Truth Value: The statement is either true (T) or false (F) 3. Negation: A statement with the opposite meaning and opposite truth value; the negation of p is ~p 4. Compound Statement:
  • 7. Vocabulary 1. Statement: A sentence that is either true or false, often represented as statements p and q 2. Truth Value: The statement is either true (T) or false (F) 3. Negation: A statement with the opposite meaning and opposite truth value; the negation of p is ~p 4. Compound Statement: Two or more statements joined by and or or
  • 8. Vocabulary 5. Conjunction: 6. Disjunction: 7. Conjunction: 8. If-Then Statements:
  • 9. Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: 7. Conjunction: 8. If-Then Statements:
  • 10. Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: A compound statement using the word or or the symbol ⋁ 7. Conjunction: 8. If-Then Statements:
  • 11. Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: A compound statement using the word or or the symbol ⋁ 7. Conjunction: A statement that fits the if-then form, providing a connection between the two phrases 8. If-Then Statements:
  • 12. Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: A compound statement using the word or or the symbol ⋁ 7. Conjunction: A statement that fits the if-then form, providing a connection between the two phrases 8. If-Then Statements: Another name for a conditional statement; in the form of if p, then q
  • 13. Vocabulary 5. Conjunction: A compound statement using the word and or the symbol ⋀ 6. Disjunction: A compound statement using the word or or the symbol ⋁ 7. Conjunction: A statement that fits the if-then form, providing a connection between the two phrases 8. If-Then Statements: Another name for a conditional statement; in the form of if p, then q p → q
  • 14. Vocabulary 9. Hypothesis: 10: Conclusion: 11. Related Conditionals: 12. Converse:
  • 15. Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: 11. Related Conditionals: 12. Converse:
  • 16. Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: The phrase that is the then part of the conditional; q 11. Related Conditionals: 12. Converse:
  • 17. Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: The phrase that is the then part of the conditional; q 11. Related Conditionals: Statements that are based off of a given conditional statement 12. Converse:
  • 18. Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: The phrase that is the then part of the conditional; q 11. Related Conditionals: Statements that are based off of a given conditional statement 12. Converse: A statement that is created by switching the hypothesis and conclusion of a conditional
  • 19. Vocabulary 9. Hypothesis: The phrase that is the if part of the conditional; p 10: Conclusion: The phrase that is the then part of the conditional; q 11. Related Conditionals: Statements that are based off of a given conditional statement 12. Converse: A statement that is created by switching the hypothesis and conclusion of a conditional q → p
  • 21. Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: 15. Logically Equivalent:
  • 22. Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: ~ p →~ q 15. Logically Equivalent:
  • 23. Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q 15. Logically Equivalent:
  • 24. Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q ~ q →~ p 15. Logically Equivalent:
  • 25. Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q ~ q →~ p 15. Logically Equivalent: Statements with the same truth values
  • 26. Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q ~ q →~ p 15. Logically Equivalent: Statements with the same truth values A conditional and its contrapositive
  • 27. Vocabulary 13. Inverse: A statement that is created by negating the hypothesis and conclusion of a conditional 14. Contrapositive: A statement that is created by switching and negating the hypothesis and conclusion of a conditional ~ p →~ q ~ q →~ p 15. Logically Equivalent: Statements with the same truth values A conditional and its contrapositive The converse and inverse of a conditional
  • 29. Vocabulary 16. Biconditional: A true compound statement that consist of the conditional and its converse
  • 30. Vocabulary 16. Biconditional: A true compound statement that consist of the conditional and its converse p ↔ q
  • 31. Vocabulary 16. Biconditional: A true compound statement that consist of the conditional and its converse p ↔ q “p if and only if q,” or “p IFF q”
  • 32. Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points a. p and q
  • 33. Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points a. p and q One meter is 100 mm and November has 30 days
  • 34. Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points a. p and q One meter is 100 mm and November has 30 days q is true, but p is false, so the conjunction is false
  • 35. Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points b. ~p ⋀ r
  • 36. Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points b. ~p ⋀ r One meter is not 100 mm and a line is defined by two points
  • 37. Example 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One meter is 100 mm q: November has 30 days r: A line is defined by two points b. ~p ⋀ r One meter is not 100 mm and a line is defined by two points Both ~p and r are true, so ~p ⋀ r is true
  • 38. Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number a. p or q AB ! "!!
  • 39. Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number a. p or q AB ! "!! is proper notation for “ray AB” or kilometers are metric units AB ! "!!
  • 40. Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number a. p or q AB ! "!! Both p and q are true, so p or q is true is proper notation for “ray AB” or kilometers are metric units AB ! "!!
  • 41. Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number b. q ⋁ r AB ! "!!
  • 42. Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number b. q ⋁ r AB ! "!! Kilometers are metric units or 15 is a prime number
  • 43. Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number b. q ⋁ r AB ! "!! Since one of the statements (q) is true, q ⋁ r is true Kilometers are metric units or 15 is a prime number
  • 44. Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number c. ~p ⋁ r AB ! "!!
  • 45. Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number c. ~p ⋁ r AB ! "!! is not proper notation for “ray AB” or 15 is a prime number AB ! "!!
  • 46. Example 2 Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “ray AB” q: Kilometers are metric units r: 15 is a prime number c. ~p ⋁ r AB ! "!! Since both ~p and r are false, ~p ⋁ r is false is not proper notation for “ray AB” or 15 is a prime number AB ! "!!
  • 47. Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
  • 48. Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
  • 49. Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
  • 50. Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
  • 51. Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis Conclusion b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game.
  • 52. Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis Conclusion b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game. Hypothesis
  • 53. Example 3 Identify the hypothesis and conclusion of each statement. a. If a polygon has eight sides, then it is an octagon. Hypothesis Conclusion b. Matt Mitarnowski will advance to the next level if he completes the Towers of Hanoi in his computer game. Hypothesis Conclusion
  • 54. Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a. Measured distance is positive.
  • 55. Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a. Measured distance is positive. Hypothesis: A distance is measured
  • 56. Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a. Measured distance is positive. Hypothesis: A distance is measured Conclusion: It is positive
  • 57. Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a. Measured distance is positive. Hypothesis: A distance is measured Conclusion: It is positive If a distance is measured, then it is positive.
  • 58. Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. b. A six-sided polygon is a hexagon
  • 59. Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. b. A six-sided polygon is a hexagon Hypothesis: A polygon has six sides
  • 60. Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. b. A six-sided polygon is a hexagon Hypothesis: A polygon has six sides Conclusion: It is a hexagon
  • 61. Example 4 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. b. A six-sided polygon is a hexagon Hypothesis: A polygon has six sides Conclusion: It is a hexagon If a polygon has six sides, then it is a hexagon.
  • 62. Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. a. If you subtract a whole number from another whole number, the result is also a whole number.
  • 63. Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. a. If you subtract a whole number from another whole number, the result is also a whole number. False
  • 64. Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. a. If you subtract a whole number from another whole number, the result is also a whole number. False 5 − 11 = −6
  • 65. Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. b. If last month was September, then this month is October.
  • 66. Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. b. If last month was September, then this month is October. True
  • 67. Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. c. When a rectangle has an obtuse angle, it is a parallelogram.
  • 68. Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. c. When a rectangle has an obtuse angle, it is a parallelogram. A rectangle cannot have an obtuse angle, so we cannot test this. All rectangles are parallelograms.
  • 69. Example 5 Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counter example. c. When a rectangle has an obtuse angle, it is a parallelogram. True: False Hypothesis A rectangle cannot have an obtuse angle, so we cannot test this. All rectangles are parallelograms.
  • 70. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse:
  • 71. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse: If MN ≅ NO, then N is the midpoint of MO.
  • 72. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse: If MN ≅ NO, then N is the midpoint of MO. False
  • 73. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse: If MN ≅ NO, then N is the midpoint of MO. False M, N, and O might not be collinear
  • 74. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Converse: If MN ≅ NO, then N is the midpoint of MO. False M, N, and O might not be collinear M N O
  • 75. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse:
  • 76. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse: If N is not the midpoint of MO, then MN ≅ NO.
  • 77. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse: False If N is not the midpoint of MO, then MN ≅ NO.
  • 78. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse: False If N is not the midpoint of MO, then MN ≅ NO. If N is not on MO, then MN could be congruent to NO.
  • 79. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Inverse: False If N is not the midpoint of MO, then MN ≅ NO. If N is not on MO, then MN could be congruent to NO. M N O
  • 80. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Contrapositive:
  • 81. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Contrapositive: If MN ≅ NO, then N is not the midpoint of MO.
  • 82. Example 6 Determine the converse, inverse, and contrapositive for the following statement. Then determine if the new statement is true. If false, give a counterexample. If N is the midpoint of MO, then MN ≅ NO. Contrapositive: True If MN ≅ NO, then N is not the midpoint of MO.