If T(x) denotes x is a trigonometric function, P(x) denotes x is a periodic function and C(x) denotes x is a continuous function then the statement “It is not the case that some trigonometric functions are not periodic” can be logically represented as
(A) ¬∃(x) [ T(x) ⋀ ¬P(x) ]
(B) ¬∃(x) [ T(x) ⋁ ¬P(x) ]
(C) ¬∃(x) [ ¬T(x) ⋀ ¬P(x) ]
(D) ¬∃(x) [ T(x) ⋀ P(x) ]
Answer: (A)
Explanation: some trigonometric functions are not periodic = ∃(x) [ T(x) ⋀ ¬P(x) ]
And it’s negation is = ¬∃(x) [ T(x) ⋀ ¬P(x) ]
Which is equivalent to “It is not the case that some trigonometric functions are not periodic” is equivalent to “All trigonometric functions are periodic” can be expression as
= ∀(x) [T(x) → P(x)] = ∀(x) [¬ T(x) ⋁ P(x)] = ∀(x) ¬ [ T(x) ⋀ ¬P(x)] = ¬∃(x) [ T(x) ⋀ ¬P(x) ]
Option (A) is correct.
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