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Logical Implication in Statement Equivalence

Understand logical implication in statement equivalence by proving the relationship between statements using logical symbols and truth tables. Learn how to show logical equivalence and implications between statements.

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Logical Implication in Statement Equivalence

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  1. VI Implication

  2. Definition • Let p and q be any two statements, p implies q if and only if p v q Ξ q • aka p implies q if the statement “p or q” is logically equivalent to the statement q • p q is read “p implies q” • p q is read “p does not imply q”

  3. Examples • p: the animal is a beagle • q: the animal is a dog • We can say p q bc The animal is a beagle or the animal is a dog is logically equivalent to The animal is a dog

  4. Prove this relationship [(p  q) ^ ~q ] ~ p To prove this we need to show [(p  q) ^ ~q ] v ~ p Ξ ~ p WHY????? p implies q if and only if p v q Ξ q

  5. Lets look it another way

  6. SO LETS PROVE IT [(p  q) ^ ~q ] v ~ p Ξ ~ p

  7. Prove another [(p  ~q) ^ q ] q

  8. HOMEWORK 5 Use a truth table to prove [ (p  q) ^ p ] q

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