# Symmetric Relation/Examples/Is a Brother of

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## Example of Symmetric Relation

Let $P$ be the set of male people.

Let $\sim$ be the relation on $P$ defined as:

- $\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { $x$ is a brother of $y$}$

Then $\sim$ is a symmetric relation.

This does not hold if $P$ is the set of *all* people.

Because if $a$ is male and $b$ are brother and sister, then:

- $a \sim b$

but:

- $b \not \sim a$

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations