Naive set theory: Relations
Properties of relations on a set
Properties of a binary relation
Properties
A binary relation \(R\) on a set \(X\) is called:
- reflexive if \(xRx\) for every \(x\in X\);
- symmetric if for all \(x,y\in X\): if \(xRy\), then \(yRx\);
- antisymmetric if for all \(x,y\in X\): if \(xRy\) and \(yRx\), then \(x=y\);
- asymmetric if there is no pair \((x,y)\in X\times X\) bsuch that \(xRy\) and \(yRx\);
- transitive if for all \(x,y,z\in X\): if \(xRy\) and \(yRz\), then \(xRz\).
Examples
The relation \(\le\) on \(\mathbb{Z}\) is reflexive, not symmetric, antisymmetric, and transitive.
The relation \(\lt\) on \(\mathbb{Z}\) is not reflexive, not symmetric, assymetric, and transitive.
The relation \(=\) on \(\mathbb{Z}\) is reflexive, symmetric, antisymmetric, and transitive.
The relation "is parallel to" on the set of lines in the plane is reflexive, symmetric, and transitive.
Let \(R\) be a relation on a set \(X\). Then:
- \(R\) is reflexive if and only if \(\mathrm{Id}(X)=\{(x,x)\mid x\in X\}\subset R\cap R^{-1}\);
- \(R\) is symmetric if and only if \(R^{-1}=R\);
- \(R\) is antisymmetrisch dan en slechts dan als \(R\cap R^{-1}\subset \mathrm{Id}_X=\bigl\{(x,x)\mid x\in X\bigr\}\);
- \(R\) is transitive if and only if \(R\circ R\subset R\).
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