Answer
a) ∀x∃y(x = 1/y)
This statement is saying that for any integer x, there exists an integer y such that x is equal to 1/y.
A counterexample to this statement would be x = 0. There does not exist any integer y such that x = 1/y, because any number divided by 0 is undefined.
b) ∀x∃y(y2 − x < 100)
This statement is saying that for any integer x, there exists an integer y such that the difference between y squared and x is less than 100.
A counterexample to this statement would be x = -100. There does not exist any integer y such that y^2 - x < 100, because y^2 will always be greater than or equal to 0, and thus, y^2 - (-100) will always be greater than or equal to 100.
c) ∀x∀y(x2 ≠ y3)
This statement is saying that for any integers x and y, x squared is not equal to y cubed.
A counterexample to this statement would be x = -1 and y = -1. (-1)^2 = 1 and (-1)^3 = -1. Therefore x^2 = y^3
Work Step by Step
a) The statement ∀x∃y(x = 1/y) is saying that for any integer x, there exists an integer y such that x is equal to 1/y. In other words, this statement is saying that any integer can be expressed as the reciprocal of another integer.
A counterexample for this statement would be x = 0. There does not exist any integer y such that x = 1/y, because any number divided by 0 is undefined. So, x = 0 is not valid value for y.
b) The statement ∀x∃y(y2 − x < 100) is saying that for any integer x, there exists an integer y such that the difference between y squared and x is less than 100. In other words, this statement is saying that for any given x, there is always an y that satisfies y^2 - x < 100.
A counterexample for this statement would be x = -100. There does not exist any integer y such that y^2 - x < 100, because y^2 will always be greater than or equal to 0, and thus, y^2 - (-100) will always be greater than or equal to 100.
c) The statement ∀x∀y(x2 ≠ y3) is saying that for any integers x and y, x squared is not equal to y cubed. In other words, this statement is saying that there is no x and y such that x^2 = y^3.
A counterexample for this statement would be x = -1 and y = -1. (-1)^2 = 1 and (-1)^3 = -1. Therefore x^2 = y^3. Since x^2 = y^3, this statement is not true for these values.
In all cases, counterexamples are values that make the statement false, in this case x = 0, x = -100, x = -1, y = -1 respectively.
