Inclusion of a Subset in Another

The inclusion of a subset in another refers to the relationship between two sets where one set is completely contained within the other. In other words, all elements of the subset are also elements of the larger set. This relationship is often denoted by the symbol ⊆, which means "is a subset of".

For example, let's consider a universal set U = {1, 2, 3, 4, 5} and two subsets A = {1, 2, 3} and B = {1, 2}. In this case, A is a subset of U because all elements of A (1, 2, and 3) are also elements of U. Similarly, B is also a subset of U because all elements of B (1 and 2) are also elements of U.

The Relationship Between Sets

The relationship between sets can be categorized into different types based on certain properties. These categories include:

1. Reflexive Relation: A relation R is reflexive if every element of a set A is related to itself. In the case of the inclusion of a subset in another, this relation is not reflexive because a set cannot be a subset of itself.

2. Symmetric Relation: A relation R is symmetric if for every element a and b in a set A, if a is related to b, then b is also related to a. In the case of the inclusion of a subset in another, this relation is not symmetric because if A is a subset of B, it does not necessarily mean that B is a subset of A.

3. Equivalence Relation: An equivalence relation is a relation that is reflexive, symmetric, and transitive. Since the inclusion of a subset in another is neither reflexive nor symmetric, it cannot be an equivalence relation.

Therefore, the correct answer to the given question is option D, which states that the inclusion of a subset in another is none of these relations.

Conclusion

The inclusion of a subset in another refers to the relationship between two sets where one set is completely contained within the other. This relationship is not reflexive, symmetric, or an equivalence relation. Hence, the correct answer is option D.