Exercise 565Logic, sets and proofs › RelationsHigh schoolAdded 3 Nov 2026

Antisymmetric relations

#antisymmetric#relation#definition

Statement

Define antisymmetry and test the relation $\subseteq$ on $\mathcal{P}(X)$ .

Hint

Antisymmetric: $a\mathcal{R}b$ and $b\mathcal{R}a$ imply $a=b$ .

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Definition. $\mathcal{R}$ antisymmetric if $(a\mathcal{R}b\land b\mathcal{R}a)\Rightarrow a=b$ .

Test. If $A\subseteq B$ and $B\subseteq A$ , then $A=B$ (extensionality). Antisymmetric.

Conclusion. $\subseteq$ is antisymmetric on $\mathcal{P}(X)$ .