Elementary Foundations: An introduction to topics in discrete mathematics

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.