Question

Determine the truth value of the statement∀x∃y(xy = 1) if the domain for the variables consists of

a) the nonzero real numbers.

b) the nonzero integers.

c) the positive real numbers

Short answer

The truth value for the statementcan be determined.

Step-by-step solution

Finding truth value for the non-zero real numbers

The real number 1/x such that \(x\left( {\frac{1}{x}} \right) = 1\)for every non-zero real number. There is \(y = 1/x\)for every non-zero real number x. The given statement is true for xy=1.

Finding truth value for the non-zero integers

The real number 1/x such that \(x\left( {\frac{1}{x}} \right) = 1\)for every non-zero real number. There is \(y = 1/x\)is not an integer and cannot have integer xy=1. The given statement is true.

Finding truth value for the positive real numbers

The real number 1/x such that \(x\left( {\frac{1}{x}} \right) = 1\)for every positive real number x. There is \(y = 1/x\)for every positive real number x. The given statement is true for xy=1.