1. Section 3.3 Using Laws of Logic

2. Using contrapositives  The negation of a hypothesis or of a conclusion is formed by denying the original hypothesis or conclusion.  Statement Symbol Negation Symbol The weather is good. P The weather is not good. ~p I will go swimming. q I will not go swimming. ~q  The inverse of the conditional statement p  q is ~p  ~q.  The contrapositive of the conditional statement p  q is ~q  ~p.

3. Examples:  Original Statement: p  q If Polly says “Hello”, then Paul says “Hello.” Hypothesis, pconclusion, q  Converse: q  p If Paul says “Hello’” then Polly says “Hello.”  Inverse: ~p  ~q If Polly does not say “Hello’” then Paul does not say “Hello.”  Contrapositive: ~q  ~p If Paul does not say “Hello’” then Polly does not say “Hello.”

4. Ex:  Statement: Tomorrow is Friday, if today is Thursday  Conditional : If today is Thursday, then tomorrow is Friday (True)  Converse : If tomorrow is Friday, then today is Thursday. (True)  Inverse : If today is not Thursday, then tomorrow is not Friday. (True)  Contrapositive : If tomorrow is not Friday, then today is not Thursday. (True)

5. Ex:  Statement: A figure is a parallelogram if it is a square  Conditional: If a figure is a square, then it is a parallelogram.(True)  Converse: If a figure is a parallelogram, then it is a square.(False)  Inverse: If a figure is not a square, then it is not a parallelogram.(False)  Contrapositive : If a figure is not a parallelogram, then it is not a square.(True)

6. Summary An important fact to remember about the contrapositive, is that it always has the SAME truth value as the original conditional statement. **If the original statement is TRUE, the contrapositive is TRUE. If the original statement is FALSE, the contrapositive is FALSE. They are said to be logically equivalent. StatementIf p, then q. Converse If q, then p. Inverse If not p, then not q. Contrapositive If not q, then not p.

7. Laws of Logical Reasoning  Law of Syllogism (like transitive property)  “If p then q, if q then r therefore if p then r”  p  q  q  r  Therefore, p  r Ex: If today is Tuesday, then I have gym. If I have gym, then I wear my sneakers. Conclusion using law of syllogism: If today is Tuesday, then I wear my sneakers.

8. Laws cont.  Law of Detachment (ordering steps to reach conclusion)  p  q  P is true  Therefore, q is true  Ex: Given <1 = 50°  A) if <1= 50°, then <2= 40°  B) if <3= 40°, then <4= 140°  C) if <4= 140°, then <5= 140°  D) <1 = 50°  E) if <2= 40°, then <3= 40°  Solution: order of steps D,A,E,B,C  Conclusion using law of detachment: <5 = 140°