Question

Determine the truth value of the statement∃x∀y(x≤y²) if the domain for the variables consists of

1. The positive real numbers
2. The integers
3. The non-zero real numbers

Short answer

The truth value for the statementcan be determined.

Step-by-step solution

Finding truth value for the positive real numbers

Let’s take exists of x value for the true statement. Take\(y = \sqrt {\frac{x}{2}} \), y is positive real number.

\(\begin{aligned}{l}{y^2} &= {\left( {\frac{{\sqrt x }}{2}} \right)^2}\\ = \frac{x}{4}\\ \le x\end{aligned}\)

There is a contradiction and noted that the y square should be less than x and not a value for true statement. The given statement is false for this domain.

Finding truth value for the integers

The integer square is non-negative and x=0 is the value for true statement. i.e., \(0 \le {y^2}.\)The given statement is true for this domain.

Finding truth value for the non-zero real numbers

The integer square is non-negative so x=0. It is the value for true statement. i.e., \(0 \le {y^2}.\)x=0 is not a domain and the value can be choose less than zero. -1 is in non-zero real number which satisfies \( - 1 \le {y^2}.\)