Geom 2point3
- 1. 2.3 Deductive Reasoning Objectives: - Use symbolic notation to represent logical statements - Form conclusions by applying the laws of logic to true statements
- 2. Using Symbolic Notation Let p can represent the hypothesis and q represent the conclusion If p then q OR p q How would we write the converse? If q, then p OR q p A biconditional can be written as If p, then q and if q, then p, OR p q
- 3. Let p be “the value of x is -5” and let q be “the absolute value of x is 5” Write p q in words. If the value of x is -5, then the absolute value of x is 5 Write q p in words. If the absolute value of x is 5, then the value of x is -5. Decide whether the biconditional statement p q is true The conditional statement is true, but the converse is false, so the biconditional is false.
- 4. Symbolic Notation for NOT Not: ~ Not p: ~p Not q: ~q The inverse of p q: ~p ~q The contrapositive of p q: ~q ~p
- 5. Example Let p be “it is raining” and q be “the soccer game is cancelled” p q ~q ~p Contrapositive: If the soccer game is not canceled, then it is not raining. Inverse: ~p ~q If it is not raining, then the soccer game is not cancelled.
- 6. Look at bottom of page 88 Review: A conditional statement is equivalent to the contrapositive Converse and inverse are equivalent Look at the car battery example at the bottom of p. 88
- 7. Using the Laws of Logic Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument . This is different from inductive reasoning, in which previous examples and patterns are used to form a conjecture.
- 8. Inductive or deductive? Andrea know that Robin is a sophomore and Todd is a junior. All the other juniors that Andrea knows are older than Robin. Therefore, Andrea reasons that Todd is older than Robin based on past observation. Andrea knows that Todd is older than Chan. She also knows that Chan is older than Robin. Andrea reasons that Todd is older than Robin based on accepted statements.
- 9. Law of Detachment If p q is a true conditional statement and p is true, then q is true.
- 10. Is this argument valid? Jamal knows that if he misses the practice the day before a game, then he will not be a starting player in the game. Jamal misses practice on Tuesday so he concludes that he will not be able to start in Wednesday’s game. Valid use of Law of Detachment
- 11. Is this argument valid? If 2 angles form a linear pair, then they are supplementary; Angle A and Angle B are supplementary, So Angle A and Angle B form a linear pair. Not a valid use of Law of Detachment 120º 60°
- 12. Law of Syllogism If p q and q r are true conditional statements, then p r is true.
- 13. Write some conditional statements about these true Zoology Facts If a bird is the fastest bird on land, then it is the largest of all birds. If a bird is the largest of all birds, then it is an ostrich. If a bird is a bee hummingbird, then it is the smallest of all birds. If a bird is the largest of all birds, then it is flightless. If a bird is the smallest bird, then it has a nest the size of a walnut half-shell.
- 14. Deductive Reasoning Over the summer, Mike visited Alabama. If Mike visits Alabama, then he will spend a day in Montgomery. If Mike spends a day in Montgomery, then he will visit the Civil Rights Memorial. Did Mike visit the Civil Rights Memorial?
- 15. Lewis Carroll’s puzzle . . . A says B lies. B says C lies. C says A and B lie. Who is telling the truth? Who is lying?
- 16. Do page 91 1 - 7 Homework: worksheets