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Logic do it! Looking Glass Through the The Looking Glass A mirror - PDF document

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IIT Bombay :: Autumn 2020 :: CS 207 :: Discrete Structures :: Manoj Prabhakaran It computers can s so easy even Logic do it! Looking Glass Through the The Looking Glass A mirror which shows the negation of every proposition Reflection

  1. IIT Bombay :: Autumn 2020 :: CS 207 :: Discrete Structures :: Manoj Prabhakaran It’ computers can s so easy even Logic do it! Looking Glass Through the

  2. The Looking Glass A mirror which shows the negation of every proposition Reflection changes T & F to F & T (resp.) ∨ & ∧ are reflected as ∧ & ∨ (resp.) Flies(Alice) ¬ Flies(Alice) is False is True Flies(Alice) ∨ ? ¬Flies(Alice) ∧ ∨ T F ∧ F T ? Flies(J’wock) ¬Flies(J’wock) T T T F F F is True is False F T F T F T ∧ F T ∨ T F F F F T T T T F T F T F

  3. The Looking Glass wire A mirror which shows the negation of every proposition Reflection changes T & F to F & T (resp.) ∨ & ∧ are reflected as ∧ & ∨ (resp.) De Morgan’ s Law ¬(p ∧ q) ≡ (¬p) ∨ (¬q) ¬(p ∨ q) ≡ (¬p) ∧ (¬q) p ¬p p ∧ q ¬p ∨ ¬q ∧ ∨ q ¬q p ¬p p ∨ q ¬p ∧ ¬q ∨ ∧ q ¬q

  4. The Looking Glass wire A mirror which shows the negation of every proposition Reflection changes T & F to F & T (resp.) ∨ & ∧ are reflected as ∧ & ∨ (resp.) p ¬p ∨ ∧ ∨ ∧ f(p,q) f’(¬p,¬q) q ¬q ¬ ∨ ¬ ∧ ¬ f(p,q) ≡ f’(¬p,¬q)

  5. The Looking Glass Reflection changes T & F to F & T (resp.) ∨ & ∧ are reflected as ∧ & ∨ (resp.) ∀ & ∃ are reflected as ∃ & ∀ (resp.) p ¬p p ∧ q ¬p ∨ ¬q ∨ ∧ q ¬q p ¬p p ∨ q ¬p ∧ ¬q ∨ ∧ q ¬q ∃ x ¬Pred(x) ∀ x Pred(x) ∃ x Pred(x) ∀ x ¬Pred(x)

  6. Two quantifiers ∃ y Likes(x,y) x y Likes(x,y) i.e., LikesSomeone(x) Alice TRUE Alice TRUE Jabberwock FALSE Flamingo TRUE Alice FALSE Jabberwock Jabberwock TRUE TRUE Flamingo FALSE Alice FALSE Jabberwock FALSE Flamingo TRUE Flamingo TRUE ∀ x ∃ y Likes(x,y) Everyone likes someone ∀ x LikesSomeone(x) True

  7. Two quantifiers ∃ y Likes(x,y) x y Likes(x,y) i.e., LikesSomeone(x) Alice TRUE Alice TRUE Jabberwock FALSE Flamingo TRUE Alice FALSE Jabberwock Jabberwock TRUE TRUE Flamingo FALSE Alice FALSE Jabberwock FALSE Flamingo TRUE Flamingo TRUE ∃ x ¬( ∃ y Likes(x,y) ) ∀ x ∃ y Likes(x,y) Everyone likes someone ∀ x LikesSomeone(x) True

  8. Two quantifiers ∃ y Likes(x,y) x y Likes(x,y) i.e., LikesSomeone(x) Alice TRUE Alice TRUE Jabberwock FALSE Flamingo TRUE Alice FALSE Jabberwock Jabberwock TRUE TRUE Flamingo FALSE Alice FALSE Jabberwock FALSE Flamingo TRUE Flamingo TRUE ∃ x ∀ y ¬Likes(x,y) ∀ x ∃ y Likes(x,y) Someone doesn’ t like anyone Everyone likes someone ∀ x LikesSomeone(x) ∃ x DoesntLikeAnyone(x) True False

  9. Two quantifiers x y Likes(x,y) Alice TRUE Alice Jabberwock FALSE Flamingo TRUE Alice FALSE Jabberwock Jabberwock TRUE Flamingo FALSE Alice FALSE Jabberwock FALSE Flamingo Flamingo TRUE ∃ y ∀ x Likes(x,y)

  10. Two quantifiers ∀ x Likes(x,y) x y Likes(x,y) i.e., EveryoneLikes(y) Alice TRUE Alice FALSE Jabberwock FALSE Flamingo FALSE Alice FALSE Jabberwock Jabberwock TRUE FALSE Flamingo FALSE Alice TRUE Jabberwock FALSE Flamingo FALSE Flamingo TRUE ∃ y ∀ x Likes(x,y) ∀ y ∃ x ¬Likes(x,y) Someone is liked by Everyone is disliked everyone by someone False True

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