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The question is:

Let $A = \{1, 2, 3, 4, 5\}$ and

Let $R = \{ (3,4), (5,5), (1,1), (2,2), (5,2), (1,4), (2,5), (3,1), (3,3), (4,1), (1,3), (4,3), (4,4)\}$ be an equivalence relation on A.

Find the [$1$] and [$3$] on $R$.

I understand what an equivalence relation is (reflexive, symmetric and transitive), but I've been trying to find what the professor wants me to do but I can't figure out what he wants by [$1$] and [$3$]. What does he mean by find [$1$] and [$3$]? Does he want me to find the sets that make up the reflexive and transitive properties?

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  • $\begingroup$ Given an equivalence relation $R$ on $A\times A$, the equivalence class of $x$ is denoted $[x]$ and is defined as: $$[x]:=\{a\in A~:~(a,x)\in R\}$$ $\endgroup$
    – JMoravitz
    Sep 29, 2017 at 17:24

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Yes, $[1]$ denotes the equivalence class of $1$, i.e., the set of all elements in $A$ that are equivalent to $1$ with respect to $R$. Hence $[1]=\{1,4,3\}$. Hope I could help :)

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What does he mean by find [$1$] and [$3$]? Does he want me to find the sets that make up the reflexive and transitive properties?

You phrase that rather awkwardly.

In general, $[a] = \{ b | (a,b) \in R \}$

In other words: $[1]$ is the set of all the objects that stand in relation $R$ to $1$.

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