Answer:

The given statement is “is equal to over the set of all rational numbers”. Here, the set of all rational numbers is denoted by Q.

To determine the options that are correct, we need to understand the meaning of the given statement.

Meaning of the given statement:

This statement refers to the equivalence relation on the set of rational numbers. An equivalence relation is a relation that satisfies the following three properties:

1. Reflexivity: a ~ a for all a in Q.

2. Symmetry: if a ~ b, then b ~ a for all a, b in Q.

3. Transitivity: if a ~ b and b ~ c, then a ~ c for all a, b, c in Q.

The equivalence class of a rational number a is the set of all rational numbers that are equivalent to a. We denote the equivalence class of a by [a].

The set of all equivalence classes of Q under the equivalence relation ~ is denoted by Q/~.

Now, let’s analyze each option:

Option A: (T)

This option is correct because the given statement refers to an equivalence relation on Q.

Option B: (S)

This option is correct because the set of equivalence classes of Q under the equivalence relation ~ is denoted by Q/~.

Option C: (R)

This option is correct because the set of all equivalence classes of Q under the equivalence relation ~ is denoted by Q/~, and this set is also known as the set of rational numbers modulo ~.

Option D: (E)

This option is correct because the set of all equivalence classes of Q under the equivalence relation ~ is denoted by Q/~, and this set is also known as the set of rational numbers modulo ~.

Therefore, options A, B, C, and D are all correct.

Set of all rational numbers is represented by the symbol, Q. Not T, S, R, E