Write the following statement in symbolic form and find its truth value: $\forall\, n \in N,\; n^2 + n$ is an even number and $n^2 - n$ is an odd number.

Write the following statement in symbolic form and find its truth value: ∀ n ∈ N, n² + n is an even number and

Question 198 of 227

\forall\, n \in N,\; n^2 + n

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2017Subjective· 2mImportance★★★★★

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Symbolize as a universally-quantified conjunction, then test both parts.

Let $p(n)$ : " $n^2+n$ is an even number", $q(n)$ : " $n^2-n$ is an odd number".

Symbolic form: $(\forall n \in N)\;[p(n) \land q(n)]$

Truth value: $n^2+n = n(n+1)$ is a product of two consecutive integers, so it is always even $\Rightarrow p(n)$ is true for all $n$ .

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