Consider a few example equivalence classes, for the specific example representative elements provided (if applicable). What other elements are in that class?
Devise a general way to describe every equivalence class, using your experience from the example classes already considered (if applicable). Make your class descriptions more meaningful than just “all elements equivalent to a specific representative element.”
Relation \(\mathord{\equiv} \) on \(A = \Z \text{,}\) where \(m \equiv n \) means \(m^2 = n^2 \text{.}\) Example equivalence classes for \(1, 10, -2, 0 \text{.}\)
Relation \(\mathord{\equiv} \) on \(A = \powset{\{a,b,c,d\}} \text{,}\) where \(X \equiv Y \) means \(\card{\cmplmnt{X}} = \card{\cmplmnt{Y}} \text{.}\) Example equivalence classes for \(\emptyset, \{a\}, \{a,b\}, \{a,b,c\}, \{a,b,c,d\} \text{.}\)
Relation \(\mathord{\equiv} \) on the vertex set \(A = V \) of a graph \(G \text{,}\) where \(v \equiv v' \) means there exists a path in \(G \) from \(v \) to \(v' \text{.}\)
Given function \(\funcdef{f}{A}{B} \text{,}\) the relation \(\mathord{\equiv} \) on the domain \(A \text{,}\) where \(a_1 \equiv a_2 \) means \(f(a_1) = f(a_2) \text{.}\)
A sequence from a set \(A \) could also be called an ordered list. For example, given distinct \(a_1,a_2 \in A \text{,}\) the finite sequences \(a_1,a_1,a_2 \) and \(a_1,a_2,a_1 \) are different sequences, because order matters in a sequence. However, as an unordered list, \(a_1,a_1,a_2 \) is the same as \(a_1,a_2,a_1 \text{.}\)
Write \(\mathscr{S}_A \) for the set of all finite sequences from \(A \text{.}\) Devise an equivalence relation \(\mathord{\equiv} \) on \(\mathscr{S}_A \) such that the quotient set \(\mathscr{S}_A / \mathord{\equiv} \) represents the set of all finite unordered lists from \(A \text{.}\)
Suppose \(\mathord{\equiv} \) and \(\mathord{\equiv}' \) are equivalence relations on a set \(A \text{.}\) Determine which of the following are also equivalence relations.
