Answer
a) ∀x∀y(x2 = y2 → x = y)
This statement is saying that for any two integers x and y, if x squared is equal to y squared, then x must be equal to y.
A counterexample to this statement would be x = -3 and y = 3.
-3^2 = 9 which is equal to 3^2 = 9.
However, x = -3 is not equal to y = 3
b) ∀x∃y(y2 = x)
This statement is saying that for any integer x, there exists an integer y such that y squared is equal to x.
A counterexample to this statement would be x = -1.
There does not exist any integer y such that y^2 = -1, as the square of any integer is always non-negative.
c) ∀x∀y(xy ≥ x)
This statement is saying that for any integers x and y, the product of x and y is greater than or equal to x.
A counterexample to this statement would be x = -1 and y = -2.
(-1)(-2) = 2 is not greater than or equal to -1.
Work Step by Step
a) The statement ∀x∀y(x2 = y2 → x = y) is saying that for any two integers x and y, if x squared is equal to y squared then x must be equal to y. In other words, this statement is saying that if two integers have the same square value, then they must be the same number.
A counterexample for this statement would be x = -3 and y = 3. -3^2 = 9 which is the same as 3^2 = 9. However, x = -3 is not equal to y = 3, so this statement is not true for these values.
b) The statement ∀x∃y(y2 = x) is saying that for any integer x, there exists an integer y such that y squared is equal to x. In other words, this statement is saying that there is always an integer y that when squared, will give the value of x.
A counterexample for this statement would be x = -1. There does not exist any integer y such that y^2 = -1, as the square of any integer is always non-negative.
c) The statement ∀x∀y(xy ≥ x) is saying that for any integers x and y, the product of x and y is greater than or equal to x. In other words, this statement is saying that the product of x and y is always greater than or equal to x.
A counterexample for this statement would be x = -1 and y = -2. (-1)(-2) = 2 is not greater than or equal to -1.
In all cases, counterexamples are values that make the statement false, in this case x = -3, y = 3, x = -1, x = -1, y = -2 respectively.
