Which of the following relations is an equivalence relation on the set of intege

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Question: Which of the following relations is an equivalence relation on the set of integers?

Options:

Correct Answer: x ~ y if x + y is even

Solution:

The relation x ~ y if x + y is even is reflexive, symmetric, and transitive, thus it is an equivalence relation.

Which of the following relations is an equivalence relation on the set of integers?

Which of the following relations is an equivalence relation on the set of integers?

Step 1: Understand what an equivalence relation is. An equivalence relation must satisfy three properties: reflexive, symmetric, and transitive.
Step 2: Define the relation. We have the relation x ~ y if x + y is even.
Step 3: Check reflexivity. For any integer x, x + x = 2x, which is even. So, x ~ x is true for all integers x.
Step 4: Check symmetry. If x ~ y, then x + y is even. This means y + x is also even (since addition is commutative). So, if x ~ y, then y ~ x.
Step 5: Check transitivity. If x ~ y and y ~ z, then x + y is even and y + z is even. This means (x + y) + (y + z) = x + 2y + z is even. Since 2y is even, x + z must also be even. Thus, if x ~ y and y ~ z, then x ~ z.
Step 6: Since the relation satisfies reflexivity, symmetry, and transitivity, we conclude that it is an equivalence relation.

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