Question

For which of these statements is x = 1, y = 2 a counterexample? (Select all that apply.) Select one or more: Y

a. For all integers x and y, is rational y/(X - 1)
b. The sum of two consecutive integers is never equal to 2
c. The difference between squares of consecutive integers is not equal to 2.
d. For all integers x and y, x2 - y2 = (x - y)^2.
e. If x and y are integers, then x^2 + y^2 + 10.

Answer 1

The statements that have a counterexample for x = 1, y = 2 are a, b, and d.

Step-by-step explanation:

A counterexample is a specific example that disproves a statement. In order to determine which statements are counterexamples, we substitute the given values of x = 1 and y = 2 into each statement and see if it holds true or not.

a. For all integers x and y, is rational y/(X - 1):

Counterexample:

2/(1-1) = 2/0, which is undefined.

b. The sum of two consecutive integers is never equal to 2:

Counterexample:

1 + 2 = 3, not equal to 2.

c. The difference between squares of consecutive integers is not equal to 2:

Counterexample:

(1^2) - (2^2) = 1 - 4 = -3, not equal to 2.

d. For all integers x and y, x^2 - y^2 = (x - y)^2:

Counterexample:

(1^2) - (2^2) = -3, not equal to (1 - 2)^2 = 1.

e. If x and y are integers, then x^2 + y^2 + 10:

Not a counterexample:

(1^2) + (2^2) + 10 = 1 + 4 + 10 = 15.

The statements that have a counterexample for x = 1, y = 2 are a, b, and d.