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Question
Write the following statement in symbolic form and find its truth value:
∀ n ∈ N, n2 + n is an even number and n2 - n is an odd number.
Solution
Let p ≡ ∀ n ∈ N, n2 + n is an even number
Let q ≡ ∀ n ∈ N, n2 - n is an odd number
The symbolic form of given statement is
(p ∧ q)
Truth value of given statement is
p ≡ ∀ n ∈ N, n2 + n is an even number (T)
q ≡ ∀ n ∈ N, n2 - n is an odd number (F)
(∵ from n = 1, n2 - n = 0, which is not an odd number)
∴ (p ∧ q) ≡ T ∧ F ≡ F
∴ the given statement is false
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