Let $R$ be the relation defined in the set $A = \{1, 2, 3, 4, 5, 6, 7\}$ by $R = \{(a, b): ext{both } a ext{ and } b ext{ are either odd or even}\}$. Show that $R$ is an equivalence relation. Further, show that all the elements of the subset $\{1, 3, 5, 7\}$ are related to

Question

Let $R$ be the relation defined in the set $A = \{1, 2, 3, 4, 5, 6, 7\}$ by $R = \{(a, b): 	ext{both } a 	ext{ and } b 	ext{ are either odd or even}\}$. Show that $R$ is an equivalence relation. Further, show that all the elements of the subset $\{1, 3, 5, 7\}$ are related to each other and all the elements of the subset $\{2, 4, 6\}$ are related to each other, but no element of the subset $\{1, 3, 5, 7\}$ is related to any element of the subset $\{2, 4, 6\}$.

Accepted Answer

> [!TLDR]
> The relation $R$ groups numbers by parity — two numbers are related if they share the same parity (both odd or both even). This partitions $A$ into two equivalence classes: odds $\{1,3,5,7\}$ and evens $\{2,4,6\}$, so $R$ is an equivalence relation.

## Why parity? The core idea

A relation is an equivalence relation if it is **reflexive**, **symmetric**, and **transitive**. Here, the rule is simple: two numbers are related when they have the same "type" — both odd or both even. That’s exactly what parity means.

This kind of relation always creates neat, non-overlapping groups (called equivalence classes). Every odd number is related to every other odd, every even to every even, and no cross-relations exist. Let’s verify each property.

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## Step-by-step verification

### 1. Reflexive: Is every element related to itself?

Take any $a \in A$. Is $a$ either odd or even? Yes — every integer is one or the other. So $a$ and $a$ are both odd or both even. Hence $(a, a) \in R$ for all $a \in A$. Reflexivity holds.

> [!TIP]
> Reflexivity is often the easiest to check — just ask: "Is an element always the same type as itself?" Here, trivially yes.

### 2. Symmetric: If $a$ is related to $b$, is $b$ related to $a$?

Suppose $(a, b) \in R$. That means $a$ and $b$ are both odd or both even. But that statement is symmetric: if $a$ and $b$ share parity, then $b$ and $a$ share parity too. So $(b, a) \in R$. Symmetry holds.

### 3. Transitive: If $a$ is related to $b$ and $b$ to $c$, is $a$ related to $c$?

Assume $(a, b) \in R$ and $(b, c) \in R$.  
- From $(a, b)$: $a$ and $b$ have the same parity.  
- From $(b, c)$: $b$ and $c$ have the same parity.

If $a$ and $b$ share parity, and $b$ and $c$ share parity, then $a$ and $c$ must share that same parity. So $(a, c) \in R$. Transitivity holds.

> [!WARNING]
> A common mistake: thinking transitivity requires all three to be the same *number*. It doesn’t — it only requires the *property* (here, parity) to be consi

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

A worked template

Why parity? The core idea

Step-by-step verification

1. Reflexive: Is every element related to itself?

2. Symmetric: If $a$ is related to $b$ , is $b$ related to $a$ ?

3. Transitive: If $a$ is related to $b$ and $b$ to $c$ , is $a$ related to $c$ ?

Method: Proving an equivalence relation and finding its equivalence classes

Steps

Common Mistakes

Why parity? The core idea

Step-by-step verification

1. Reflexive: Is every element related to itself?

2. Symmetric: If aaa is related to bbb, is bbb related to aaa?

3. Transitive: If aaa is related to bbb and bbb to ccc, is aaa related to ccc?

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2. Symmetric: If $a$ is related to $b$ , is $b$ related to $a$ ?

3. Transitive: If $a$ is related to $b$ and $b$ to $c$ , is $a$ related to $c$ ?

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