Final answer:
The truth value of the statement ∃x∀y(x ≤ y^2) for the domain of positive real numbers is true. This is because for any positive x, we can find a y where y^2 is greater than or equal to x.
Step-by-step explanation:
The student is asking to determine the truth value of the logical mathematical statement ∃x∀y(x ≤ y2) with the domain specified as the positive real numbers. To do this, let’s break down the statement: it asserts there exists a number x such that for all numbers y in the positive real numbers, x is less than or equal to y squared.
If we pick x = 0 (which is not a positive real number), then the statement is clearly true for any positive y, because 0 will always be less than or equal to y2. However, since 0 is not included in the domain of positive real numbers, we must choose a positive x. Consider the smallest possible positive real number ε (epsilon). For a sufficiently small ε, there exists a y, namely the square root of ε, such that y2 = ε. But as y increases, y2 will eventually be greater than ε. Thus, for every number x > 0, there is a y such that x ≤ y2. Consequently, the statement is true for the positive real numbers.