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how to prove equivalence relations in set theory?

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You only need to show 3 properties, of which 2 usually are obvious. I don't see why you think this is tedious
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Yes, you need to prove all three properties. As a worked example, let n be an integer, and define a~b iff a-b is an integer multiple of n. This satisfies:

1. Reflexivity: a~a because a-a=0=0n, which is an integer multiple of n.

2. Symmetry: If a-b=kn for some integer k, then b-a=-kn, which is also an integer multiple of n.

3. Transitivity: If a-b=kn and b-c=rn for some integers k and r, then a-c = (a-b)+(b-c) = kn+rn = (k+r)n, which is an integer multiple of n.

I don't think those were that tedious (1 and 2 were practically automatic), and I don't think this example is all that much easier than your average. Instead of proving these conditions, you could instead prove that every integer belongs to one and only one equivalence class, which is an equivalent condition, but that's rarely so much faster or easier.
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