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Question
#### Which one of the following is false? Read ^ as AND, v as OR, ~ as NOT, -> as one way implication and <-> as two way implication.

Answer

((x v y) <-> (~x -> ~y))

= (x v y) <-> (~ (~x) v ~y)

= (x v y) <-> (x v ~y)

= ((x v y) ^ (x v ~y)) v (~(x v y) ^ ~(x v ~y))

= ((x v (y ^ ~y)) v ((~x ^ ~y) ^ (~x ^ y))

= (x v F) v ((~x ^ ~y) ^ (~x ^ y))

= x v (~x ^ (y ^ ~y)) = x v (~x ^ F)

= x v F

=x