Exercise 274Logic, sets and proofs › RelationsHigh schoolAdded 3 Nov 2026

Symmetric relations

#symmetric#relation#definition

Statement

Define a symmetric relation and determine whether $a\mathcal{R}b\iff a\le b$ on $\mathbb{R}$ is symmetric.

Hint

Symmetric: $a\mathcal{R}b\Rightarrow b\mathcal{R}a$ .

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Definition. $\mathcal{R}$ is symmetric if $\forall a,b$ , $a\mathcal{R}b\Rightarrow b\mathcal{R}a$ .

Test. $1\le2$ but $2\le1$ is false. Not symmetric.

Conclusion. $\le$ is not symmetric (it is antisymmetric).