List the members of the equivalence relation on the set {1, 2, 3, 4} defined by the given partition { {1}, {2, 4}, {3} }. Also, find the equivalence classes [1], [2], [3], and [4].

To determine the members of the equivalence relation, we need to understand what an equivalence relation and a partition mean in this context.

An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive. Given a set S, a partition of S is a collection of non-empty subsets of S that cover all elements of S, and no two subsets in the collection have any common elements.

In this case, the set is {1, 2, 3, 4}, and the given partition is { {1}, {2, 4}, {3} }.

To determine the members of the equivalence relation, we need to consider the partition subsets.

1. {1}:
Since {1} is a subset of S, it is one of the equivalence classes for the equivalence relation. Thus, [1] = {1}.

2. {2, 4}:
Elements 2 and 4 are in the same subset of the partition, so they are equivalent. Thus, [2] = [4] = {2, 4}.

3. {3}:
Since {3} is a subset of S, it is one of the equivalence classes for the equivalence relation. Thus, [3] = {3}.

To summarize, the members of the equivalence relation on the set {1, 2, 3, 4} defined by the given partition { {1}, {2, 4}, {3} } are as follows:

Equivalence class [1] = {1}
Equivalence class [2] = {2, 4}
Equivalence class [3] = {3}

Note that the equivalence relation partitions the set into mutually exclusive subsets such that every element is in exactly one subset.