Consider the relation R = {(x, y) : x, y ∈ A, x = y} defined on the set A = {1, 2, 3, 4}.
(i) Show that R is an equivalence relation. (2)
(ii) Hence write the equivalence classes. (1)

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> [!TLDR]
> R = {(x,y): x = y} is just the equality relation on A, which is always reflexive, symmetric and transitive — so it is an equivalence relation, and each element forms its own singleton equivalence class.
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> **(i) R is an equivalence relation.** Here $A=\{1,2,3,4\}$ and $R=\{(x,y): x,y\in A,\ x=y\} = \{(1,1),(2,2),(3,3),(4,4)\}$.
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> *Reflexive:* For every $x\in A$, $x=x$ is true, so $(x,x)\in R$ for all $x\in A$. Hence R is reflexive.
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> *Symmetric:* If $(x,y)\in R$ then $x=y$. Since equality is symmetric, $y=x$, so $(y,x)\in R$. Hence R is symmetric.
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> *Transitive:* If $(x,y)\in R$ and $(y,z)\in R$, then $x=y$ and $y=z$, so $x=z$, i.e. $(x,z)\in R$. Hence R is transitive.
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> Since R is reflexive, symmetric and transitive, R is an equivalence relation on A.
> …

Q.Consider the relation R = {(x, y) : x, y ∈ A, x = y} defined on the set A = {1, 2, 3, 4}.

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Q.Consider the relation R = {(x, y) : x, y ∈ A, x = y} defined on the set A = {1, 2, 3, 4}.

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