Section 17.2 Operations on relations

Consider the relations \(\mathord{\lt} \) and \(\mathord{=} \) on \(\R \text{,}\) and let \(R \) be the union \(\mathord{\lt} \cup \mathord{=} \text{.}\) Then \(x \mathrel{R} y \) means that at least one of \(x \lt y \) or \(x = y \) is true. That is, \(R \) is the same as the relation \(\mathord{\le} \text{.}\)

Let \(U \) be a universal set and consider the relation \(\mathord{\subseteq} \) on \(\powset{U} \text{.}\) Then \(A \mathrel{\cmplmnt{\mathord{\subseteq}}} B \) means that \(A \) is not a subset of \(B \text{,}\) which can only happen if some elements of \(A \) are not in \(B \text{.}\) In other words, \(A \mathrel{\cmplmnt{\mathord{\subseteq}}} B \) means that \(A \cap \cmplmnt{B} \ne \varnothing \text{.}\)

Let \(H \) represent the set of all living humans, and let \(R \) represent the relation on \(H \) where \(h_1 \mathrel{R} h_2 \) means human \(h_1 \) is the parent of human \(h_2 \text{.}\) Then \(h_2 \mathrel{\inv{R}} h_1 \) means human \(h_2 \) is the child of human \(h_1 \text{.}\) Both relations express the same information, but in a different order.

Recall that \(\mid \) is a relation on \(\posset{\N} \) where \(m \mid n \) means that \(m \) divides \(n \text{.}\) Then for the inverse relation, \(n \inv{\mid} m \) means \(n \) is a multiple of \(m \text{.}\) Both relations express the same information, but in a different order.

Let \(\mathscr{L} \) represent the set of all possible logical statements. We have a relation \(\equiv \) on \(\mathscr{L} \text{,}\) where \(A \equiv B \) means that logical statement \(A \) involves the same statement variables and has the same truth table as logical statement \(B \text{.}\) Since \(A \equiv B \) if and only if \(B \equiv A \text{,}\) we conclude that the logical equivalence relation on \(\mathscr{L} \) is its own inverse.