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Section 3.3

Section 3.3 . Using Laws of Logic. Contrapositives. The negation of a hypothesis or of a conclusion is formed by denying the original hypothesis or conclusion .

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Section 3.3

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  1. Section 3.3 Using Laws of Logic

  2. Contrapositives • The negation of a hypothesisor of a conclusion is formed by denying the original hypothesis or conclusion. • Statement Symbol Negation SymbolThe weather is good. P The weather is not good. ~pI will go swimming. q I will not go swimming. ~q • The inverse of the conditional statement p  q is ~p ~q. • Thecontrapositiveof the conditional statement p  q is ~q  ~p.

  3. Examples: • Original Statement: p q If Polly says “Hello”, then Paul says “Hello.” Hypothesis, p conclusion, q • Converse: q  p If Paul says “Hello’” then Polly says “Hello.” • Inverse:~p  ~qIf Polly does not say “Hello’” then Paul does not say “Hello.” • Contrapositive: ~q  ~pIf 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 **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.

  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(orderingsteps 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°

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