1.3.1 Conditional Statements
- 1. Conditional Statements The student is able to (I can): • Identify, write, and analyze conditional statements. • Write the inverse, converse, and contrapositive of a conditional statement. • Write a counterexample to a false conjecture.
- 2. conditionalconditionalconditionalconditional statementstatementstatementstatement – a statement that can be written as an “if-then” statement. Example: IfIfIfIf today is Saturday, thenthenthenthen we don’t have to go to school. hypothesishypothesishypothesishypothesis – the part of the conditional followingfollowingfollowingfollowing the word “if” (underline once). “today is Saturday” is the hypothesis. conclusionconclusionconclusionconclusion – the part of the conditional followingfollowingfollowingfollowing the word “then” (underline twice). “we don’t have to go to school” is the conclusion.
- 3. Examples NotationNotationNotationNotation Conditional statement: p → q, where p is the hypothesis and q is the conclusion. Identify the hypothesis and conclusion: 1. If I want to buy a book, then I need some money. 2. If today is Thursday, then tomorrow is Friday. 3. Call your parents if you are running late.
- 4. Examples NotationNotationNotationNotation Conditional statement: p → q, where p is the hypothesis and q is the conclusion. Identify the hypothesis and conclusion: 1. If I want to buy a book, then I need some money. 2. If today is Thursday, then tomorrow is Friday. 3. Call your parents if you are running late.
- 5. Examples To write a statement as a conditional, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other. Write a conditional statement: • Two angles that are complementary are acute. • Even numbers are divisible by 2.
- 6. Examples To write a statement as a conditional, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other. Write a conditional statement: • Two angles that are complementary are acute. If two angles are complementary, then they are acute. • Even numbers are divisible by 2. If a number is even, then it is divisible by 2.
- 7. To prove a conjecture false, you just have to come up with a counterexample. • The hypothesis must be the samesamesamesame as the conjecture’s and the conclusion is differentdifferentdifferentdifferent. Example: Write a counterexample to the statement, “If a quadrilateral has four right angles, then it is a square.”
- 8. To prove a conjecture false, you just have to come up with a counterexample. • The hypothesis must be the samesamesamesame as the conjecture’s and the conclusion is differentdifferentdifferentdifferent. Example: Write a counterexample to the statement, “If a quadrilateral has four right angles, then it is a square.” A counterexample would be a quadrilateral that has four right angles (true hypothesis) but is not a square (different conclusion). So a rectanglerectanglerectanglerectangle would work.
- 9. Examples Each of the conjectures is false. What would be a counterexample? If I get presents, then today is my birthday. If Lamar is playing football tonight, then today is Friday.
- 10. Examples Each of the conjectures is false. What would be a counterexample? If I get presents, then today is my birthday. • A counterexample would be a day that I get presents (true hyp.) that isn’t my birthday (different conc.), such as Christmas. If Lamar is playing football tonight, then today is Friday. • Lamar plays football (true hyp.) on days other than Friday (diff. conc.), such as games on Thursday.
- 11. Examples Determine if each conditional is true. If false, give a counterexample. 1. If your zip code is 76012, then you live in Texas. TrueTrueTrueTrue 2. If a month has 28 days, then it is February. September also has 28 days, which proves the conditional false. Texas 76012
- 12. negation ofnegation ofnegation ofnegation of pppp – “Not p” Notation: ~p Example: The negation of the statement “Blue is my favorite color,” is “Blue is notnotnotnot my favorite color.” Related ConditionalsRelated ConditionalsRelated ConditionalsRelated Conditionals SymbolsSymbolsSymbolsSymbols Conditional p → q Converse q → p Inverse ~p → ~q Contrapositive ~q →~p
- 13. Example: Write the conditional, converse, inverse, and contrapositive of the statement: “A cat is an animal with four paws.” TypeTypeTypeType StatementStatementStatementStatement Conditional (p → q) If an animal is a cat, then it has four paws. Converse (q → p) If an animal has four paws, then it is a cat. Inverse (~p → ~q) If an animal is not a cat, then it does not have four paws. Contrapositive (~q → ~p) If an animal does not have four paws, then it is not a cat.
- 14. Example: Write the conditional, converse, inverse, and contrapositive of the statement: “When n2 = 144, n = 12.” TypeTypeTypeType StatementStatementStatementStatement Truth ValueTruth ValueTruth ValueTruth Value Conditional (p → q) If n2 = 144, then n = 12. F (n = –12) Converse (q → p) If n = 12, then n2 = 144. T Inverse (~p → ~q) If n2 ≠ 144, then n ≠ 12 T Contrapositive (~q → ~p) If n ≠ 12, then n2 ≠ 144 F (n = –12)
- 15. biconditionalbiconditionalbiconditionalbiconditional – a statement whose conditional and converse are both true. It is written as “pppp if and only ifif and only ifif and only ifif and only if qqqq”, “pppp iffiffiffiff qqqq”, or “pppp ↔↔↔↔ qqqq”. To write the conditional statement and converse within the biconditional, first identify the hypothesis and conclusion, then write p → q and q → p. A solution is a base iff it has a pH greater than 7. p → q: If a solution is a base, then it has a pH greater than 7. q → p: If a solution has a pH greater than 7, then it is a base.
- 16. Writing a biconditional statement: 1. Identify the hypothesis and conclusion. 2. Write the hypothesis, “if and only if”, and the conclusion. Example: Write the converse and biconditional from: If 4x + 3 = 11, then x = 2. Converse: If x = 2, then 4x + 3 = 11. Biconditional: 4x + 3 = 11 iff x = 2.