Opening the book…
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It is often useful to modify or combine relations. In Proposition 2.5.1, we considered the union of relations, which is just the union of two relations considered as sets of pairs. Similarly, in Proposition 2.5.2, we considered the relative difference of relations. Here are some other operations we can perform on relations.
Let \(R\), \(S\) be relations, and \(A\) be any set.
The inverse of \(R\) is \(R^{-1} = \Setabs{\tuple{y, x}}{\tuple{x, y} \in R}\).
The relative product of \(R\) and \(S\) is \((R \mid S) = \{\tuple{x, z} : \exists y(Rxy \land Syz)\}\).
The restriction of \(R\) to \(A\) is \(\funrestrictionto{R}{A}= R \cap A^2\).
The application of \(R\) to \(A\) is \(\funimage{R}{A} = \{y : (\exists x \in A)Rxy\}\)
Let \(S \subseteq \Int^2\) be the successor relation on \(\Int\), i.e., \(S = \Setabs{\tuple{x, y} \in \Int^2}{x + 1 = y}\), so that \(Sxy\) iff \(x + 1 = y\).
\(S^{-1}\) is the predecessor relation on \(\Int\), i.e., \(\Setabs{\tuple{x,y}\in\Int^2}{x -1 =y}\).
\(S\mid S\) is \(\Setabs{\tuple{x,y}\in\Int^2}{x + 2 =y}\)
\(\funrestrictionto{S}{\Nat}\) is the successor relation on \(\Nat\).
\(\funimage{S}{\{1,2,3\}}\) is \(\{2, 3, 4\}\).
Let \(R \subseteq A^2\) be a binary relation.
The transitive closure of \(R\) is \(R^+ = \bigcup_{0 < n \in \Nat} R^n\), where we recursively define \(R^1 = R\) and \(R^{n+1} = R^n \mid R\).
The reflexive transitive closure of \(R\) is \(R^* = R^+ \cup \Id{X}\).
Take the successor relation \(S \subseteq \Int^2\). \(S^2xy\) iff \(x + 2 = y\), \(S^3xy\) iff \(x + 3 = y\), etc. So \(S^+xy\) iff \(x + n = y\) for some \(n > 1\). In other words, \(S^+xy\) iff \(x < y\), and \(S^*xy\) iff \(x \le y\).
Show that the transitive closure of \(R\) is in fact transitive.
This page titled 1.2.7: Operations on Relations was last modified on Fri, 08 Mar 2024 05:56:12 GMT and is shared under a CC BY license and was authored, remixed, and/or curated by Richard Zach et al. (Open Logic Project) .
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