Question

To determine the interpretation of the equivalence classes for the equivalence relation.

Short answer

The equivalence classes of \((a,b)\) is \(\{ (x,y)\mid x - y = a - b\} \).

Step-by-step solution

Given data

The equivalence class \((a,b)\) is described by the integer \((a - b)\).

Formula used of equivalence relation

Let\(R\)is an equivalence relation defined on set\(S\)such that\(s \in S\).

\((s) = \{ x \in S\mid (s,x) \in R\} \)is known as an equivalence class of\(s\).

The equivalence relation is\(R = \{ (a,b),(c,d)\mid a + d = b + c\} \)on the set of ordered pairs of positive integers.

Find equivalence classes

The equivalence class \((a,b)\) is described by the integer \((a - b)\). This can be negative, positive or zero.

For any \((a,b) \in {Z^ + } \times {Z^ + }\)

Now we find equivalence class.

\(\begin{array}{c}((a,b)) = \{ (x,y)\mid ((a,b),(x,y)) \in R\} \\ = \{ (x,y)\mid a + y = b + x\} \\ = \{ (x,y)\mid x - y = a - b\} \end{array}\)

Hence, the equivalence classes of \((a,b)\) is \(\{ (x,y)\mid x - y = a - b\} \).