Symmetric Relation Example:

A symmetric relation on a set $A$ satisfies the property that for any $a, b \in A$, if $(a, b) \in R$, then $(b, a) \in R$.

Example: Let $A = \{1, 2, 3\}$ and define a relation $R$ on $A$ as: $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$

This is symmetric because for every pair $(a, b) \in R$, the pair $(b, a)$ is also in $R$. Specifically, since $(1, 2) \in R$, we also have $(2, 1) \in R$, which confirms the symmetry.

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Antisymmetric Relation Example:

An antisymmetric relation on a set $A$ satisfies the property that for any $a, b \in A$, if $(a, b) \in R$ and $(b, a) \in R$, then $a = b$.

Example: Let $A = \{1, 2, 3\}$ and define a relation $R$ on $A$ as: $R = \{(1, 1), (2, 2), (3, 3), (1, 2)\}$

This is antisymmetric because if $(a, b) \in R$ and $(b, a) \in R$, it implies $a = b$. In this case, $(1, 2) \in R$ but $(2, 1) \notin R$, so the condition holds, and the relation is antisymmetric.

Would you like more details or examples?

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5 Related Questions:

1. Can a relation be both symmetric and antisymmetric? Why or why not?
2. How do symmetric and antisymmetric relations apply to real-world situations?
3. What is the difference between reflexive and symmetric relations?
4. Can you provide examples of reflexive and antisymmetric relations?
5. What are some common mathematical structures that use symmetric and antisymmetric relations?

Tip: Symmetric relations often appear in undirected graphs, where edges have no direction, while antisymmetric relations are used in structures like orders, where direction matters.