# Function Relation (Discrete math) Let A=\{0,1,2\} and r=\{(0,0),(1,1),(2,2)\}

Function Relation (Discrete math)
Let $$\displaystyle{A}={\left\lbrace{0},{1},{2}\right\rbrace}$$ and $$\displaystyle{r}={\left\lbrace{\left({0},{0}\right)},{\left({1},{1}\right)},{\left({2},{2}\right)}\right\rbrace}$$
Show that r is an equivalence relation on A.

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Step 1
The relation is reflexive, since $$\displaystyle{\left({a},{a}\right)}\in{R}$$, where $$\displaystyle{a}\in{A}$$
Also, since $$\displaystyle{a}={b}$$ where $$\displaystyle{\left({a},{b}\right)}\in{R}$$ therefore, the relation is symmetric.
Step 2
The relation is trivial transitive, as
$$\displaystyle{\left({a},{b}\right)}\in{R},{\left({b},{c}\right)}\in{R},\rightarrow{\left({a},{c}\right)}\in{R}$$
$$\displaystyle{a}={b}={c}$$
The relation is reflexive, symmetric and transitive.
therefore, r is an equivalence relation.