Answer:

To determine which statement is true, we need to analyze the properties of equivalence relations and apply them to the given relations R and S.

Properties of Equivalence Relations:

1. Reflexivity: Every element of the set A is related to itself.
2. Symmetry: If two elements a and b are related, then b and a are also related.
3. Transitivity: If two elements a and b are related, and b and c are related, then a and c are also related.

Statement a: R ∩ S and R ∩ S' are both equivalence relations.

If R ∩ S is an equivalence relation, it must satisfy the three properties stated above. However, the intersection of two equivalence relations does not necessarily result in another equivalence relation. For example, consider the following counterexample:

Let A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3)} (equivalence relation)
Let S = {(2, 2), (3, 3)} (equivalence relation)

The intersection of R and S is R ∩ S = {(2, 2), (3, 3)} which satisfies reflexivity, symmetry, and transitivity. However, the intersection of R and S' (complement of S) is R ∩ S' = {(1, 1), (2, 2), (3, 3)} which does not satisfy symmetry.

Therefore, statement a is not always true.

Statement b: R ∪ S is an equivalence relation.

The union of two equivalence relations also does not necessarily result in another equivalence relation. For example, consider the following counterexample:

Let A = {1, 2, 3} and R = {(1, 1), (2, 2)} (equivalence relation)
Let S = {(2, 2), (3, 3)} (equivalence relation)

The union of R and S is R ∪ S = {(1, 1), (2, 2), (3, 3)} which satisfies reflexivity and symmetry, but it does not satisfy transitivity.

Therefore, statement b is not always true.

Statement c: R ∩ S is an equivalence relation.

The intersection of two equivalence relations does satisfy the properties of an equivalence relation. Let's analyze each property:

1. Reflexivity: Since R and S are equivalence relations, every element of A is related to itself in both R and S. Therefore, every element of A is related to itself in R ∩ S.

2. Symmetry: If two elements a and b are related in both R and S, then b and a are also related in both R and S. Therefore, b and a are related in R ∩ S.

3. Transitivity: If two elements a and b are related in both R and S, and b and c are related in both R and S, then a and c are also related in both R and S. Therefore, a and c are related in R ∩ S.

Since R ∩ S satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.

Therefore, statement c is true.

Statement d: Neither R ∪ S nor R ∩ S is an equivalence relation