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)
⟺(~(p⋀~q)⋁r) [∵De morgan’s law]
⟺(p⋀~q)→r
∴(p→(q⋁r) )⟺((p⋀~q)→r)
Option (b);
(p→r)⋀(q→r)⟺(~p⋁r)⋀(~q⋁r)
⟺(~p⋀~q)⋁r [∵ De morgan’s law]
⟺~(p⋁q)⋁r
⟺(p⋁q)→r
∴(p→r)⋀(q→r)⟺(p⋁q)→r
Option (c):
p→(q→r)⟺~p⋁(q→r)
⟺ ~p⋁(~q⋁r)
⟺~q⋁(~p⋁r)
⟺~q⋁(p→r)
⟺q→(p→r)
Option (d):
(p↔q)⟺(p→q)⋀(q→p)
⟺(~p⋁q)⋀(~q⋁p)
⟺(~p⋁q)⋀(p⋁~q)
⟺(~p⋀~q)⋁(p⋀q)
⟺ ~(p⋁q)⋁(p⋀q)
(p↔q)⟺(p⋁q)⋁(p⋀q)is wrong.


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