Question

a)Suppose that a statement of the form\(\forall xP \left( x \right)\)is false. How can this be proved?

b)Show that the statement “For every positive integer\(n,{n^2} \ge 2n\)” is false.

Short answer

(a) The statement is false.

(b) The statement is false.

Step-by-step solution

Introduction

A statement is false if it does not satisfies the given condition in the statement.

(a) Proof of falsity

Suppose that\(P \left( x \right)\)is “\({x^2} > 0\)”

Then the statement\(\forall xP \left( x \right)\)is false, where the universe of discourse consists of all integers, give a counter example.

See that \(x = 0\)is a counter example because\({x^2} = 0\)

Where\(x = 0\), so that\({x^2}\)is not greater than 0 when\(x = 0\).

(b) Proof of falsity

Let\(P \left( x \right)\)be the statement that “For every positive integral value\(x\),\({x^2} \ge 2x\)” is a counterexample.

If\(x = 1\), then the statement\(P \left( 1 \right)\)is false because\({1^2} = 1\)is not greater than equal to 2.

Therefore, the statement “For every positive integral value\(n,{n^2} \ge 2n\)is false.