GATE | GATE 2017 MOCK II | Question 28

Which of the following argument is invalid ?

(A) (p→(q⋁r) )⟺((p⋀~q)→r)
(B) ((p→r)⋀(q→r)⟺((p⋁q)→r)
(C) p→(q→r)⟺q→(p→r)
(D) (p↔q)⟺(p⋁q)⋁(p⋀q)

Answer: (D)

Explanation: “⟺” mean equal.
ifA⟺B means A is equal to B.
Option (a);
(p→(q⋁r) )⟺(~p⋁(q⋁r) )
⟺(~(p⋀~q)⋁r) [∵De morgan’s law]
∴(p→(q⋁r) )⟺((p⋀~q)→r)
Option (b);
⟺(~p⋀~q)⋁r [∵ De morgan’s law]
Option (c):
⟺ ~p⋁(~q⋁r)
Option (d):
⟺ ~(p⋁q)⋁(p⋀q)
(p↔q)⟺(p⋁q)⋁(p⋀q)is wrong.

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