Question

Determine whether these are valid arguments.

a) If x is a positive real number, then \({x^2}\) is a positive real number. Therefore, if \({a^2}\) is positive, where a is a real number, then a is a positive real number.

b) If \({x^2} \ne 0\), where x is a real number, then\(x \ne 0\). Let a be a real number with \({a^2} \ne 0\); then \(a \ne 0\).

Short answer

1. The argument is not valid.
2. The argument is valid.

Step-by-step solution

Part (a)

Let \(P(x)\) be the proposition that “x is a positive real number.”

Let \(Q(x)\) be the proposition that “\({x^2}\)is a positive real number.”

Given \(\forall x (P(x) \to Q(x))\)

By Universal instantiation, \(P(a) \to Q(a)\), that is, “If ais a positive real number then\({a^2}\) is also a positive real number.” But \(Q(a) \to P(a)\)cannot be concluded by this proposition.

Therefore, if \({a^2}\)is a positive real number then aneed not be a positive real number.

Hence this argument is not valid.

Part (b)

Let \(P(x)\) be the proposition that \({x^2} \ne 0\).

Let \(Q(x)\) be the proposition that \(x \ne 0\).

Given \(\forall x (P(x) \to Q(x))\)

By Universal instantiation, \(P(a) \to Q(a)\), that is, “If \({a^2} \ne 0\) then\(a \ne 0\).” where is a positive real number.

Therefore, this argument is valid.