If $R=\{(2,3),(3,4),(3,3)\}$ is a relation on the set $A=\{2,3,4\}$, then find the elements, those should be added to $R$, to make it an equivalence relation.

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Accepted Answer

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> [!TLDR]
> Add reflexive pairs, symmetric pairs, and then close under transitivity; the result is the full relation $A	imes A$.
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> $A=\{2,3,4\}$, $R=\{(2,3),(3,4),(3,3)\}$.
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> **Step 1 — Reflexivity:** need $(2,2),(3,3),(4,4)$. $(3,3)$ is already present; add $(2,2)$ and $(4,4)$.
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> **Step 2 — Symmetry:** for $(2,3)\in R$, need $(3,2)$; for $(3,4)\in R$, need $(4,3)$. Add $(3,2)$ and $(4,3)$.
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> **Step 3 — Transitivity check with the new pairs:**
> - $(2,3)$ and $(3,4)\in R \Rightarrow$ need $(2,4)$. Add $(2,4)$, and then by symmetry also $(4,2)$.
> - Checking all further combinations (e.g. $(3,2)\&(2,4)\Rightarrow(3,4)$ ✓ already there; $(4,3)\&(3,2)\Rightarrow(4,2)$ ✓ already added) shows no more pairs are forced.
> …

Q.If $R=\{(2,3),(3,4),(3,3)\}$ is a relation on the set $A=\{2,3,4\}$ , then find the elements, those should be added to $R$ , to make it an equivalence relation.

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

A worked template

Q.If R={(2,3),(3,4),(3,3)}R=\{(2,3),(3,4),(3,3)\}R={(2,3),(3,4),(3,3)} is a relation on the set A={2,3,4}A=\{2,3,4\}A={2,3,4}, then find the elements, those should be added to RRR, to make it an equivalence relation.

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Q.If $R=\{(2,3),(3,4),(3,3)\}$ is a relation on the set $A=\{2,3,4\}$ , then find the elements, those should be added to $R$ , to make it an equivalence relation.

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