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Which of the relations are equivalence relations? For each equivalence relation, describe the associated equivalence classes.

1) $A=\mathbb{Q} ,\ x \operatorname{R} y \text{ iff } |x| \leqslant |y|$

2) $A=\mathbb{Z} ,\ x \operatorname R y \text{ iff }x-y\text{ is a multiple of }3$

3) $A=\{0,1,2,3,4,5\} ,\ x \operatorname{R} y\text{ iff }x+y=5$

4) $A=\mathbb{N},\ x \operatorname R y\text{ iff }x$ is odd.

This is the question my professor gave. From my understanding 1 is not in equivalence relation because it is reflexive, transitive but not symmetric.

For 2) I know its in equivalence relation because a)Reflexive: $x-x=0$, $0= 3\times0$.

b)Symmetric: $x-y=3k$; $y-x = -3k \implies y - x = 3(-k) \implies y - x = 3K$.

c) Transitive: $x - y = 3k$; $y - z = 3i$; $(x-y)+(y-z) = 3k + 3i \implies x - z = 3 (k+i) \implies x - z = 3K$.

But idk about 3rd and 4th and not sure about equivalence class, can someone help me?

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You are right about 1) and 2).

3) $R$ is not reflexive, since $0\not\operatorname{R}0$. Therefore, $R$ is not an equivalence relation.

4) $R$ is not reflexive, since $2\not\operatorname{R}2$. Therefore, $R$ is not an equivalence relation.

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