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Understanding Contrapositives and Laws of Logic in Conditional Statements

This section explores the concepts of contrapositives, inverses, and converses in logical reasoning. We define how to negate hypotheses and conclusions, providing examples for clarity. It emphasizes the logical equivalence of statements and their contrapositives. Additionally, we introduce the Law of Syllogism and the Law of Detachment, demonstrating their application in reaching conclusions. Understanding these laws is essential for logical reasoning and accurate deduction, making this guide invaluable for students and professionals alike.

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Understanding Contrapositives and Laws of Logic in Conditional Statements

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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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