Question

Any subset of A × A is called a relation on the set A. A relation R on A is symmetric if
(a, b) ∈ R ⇒ (b, a) ∈ R ∀ a, b ∈ A. Give one example each, with justification, of
i) a symmetric relation on ,
ii) a relation that is not symmetric on the set {2, 3, 5, 7}.

Answer 1

Answer on Question #44535 - Math - Discrete Mathematics

Any subset of $A \times A$ is called a relation on the set $A$. A relation $R$ on $A$ is symmetric if $(a, b) \in R \Rightarrow (b, a) \in R \ \forall a, b \in A$. Give one example each, with justification, of

i) a symmetric relation on,

ii) a relation that is not symmetric on the set $[2,3,5,7]$.

Solution.

i) According to the definition, if $R$ contains an ordered pair $(a, b)$, it also contains an ordered pair $(b, a)$.

For example:

$$R = \{(2,3), (3,2), (3,7), (7,3), (5,5)\} \quad \text{- all the pairs are symmetric}.$$

ii) Using the definition, we can build a relation that is not symmetric on the given set.

$$R = \{(2,3), (3,5), (5,3), (7,7)\} \quad \text{- relation contains the pair } (2,3), \text{ but it doesn't contain symmetric pair } (3,2).$$

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