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

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

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

Steps: 6

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.