Q-4. Suppose A is the set composed of all ordered pairs of positive integers. Let R be the relation defined on A where (a, b) R (c, d) means that a+d=b+c.

Prove that R is an equivalence relation.

Find [ (2,4) ].

Step-by-step explanation:

REcall that given a set A, * is a equivalence relation over A if

- for a in A, then a*a.

- for a, b in A. If a*b, then b*a.

- for a, b, c in A. If a*b and b*c then a*c.

Consider A the set of all ordered pairs of positive integers.

- Let (a, b) in A. Then a+b = a+b. So, by definition (a, b) R (a, b).

- Let (a, b), (c, d) in A and suppose that (a, b) R (c, d). Then, by definition a+d = b+c. Since the + is commutative over the integers, this implies that d+a = c+b. Then (c, d) R (a, b).

- Let (a, b), (c, d), (e, f) in A and suppose that (a, b) R (c, d) and (c, d) R (e, f). Then

a+d = b+c, c+f = d+e. We have that f = d+e-c. So a+f = a+d+e-c. From the first equation we find that a+d-c = b. Then a+f = b+e. So, by definition (a, b) R (e, f).

So R is an equivalence relation.

[ (a, b) ] is the equivalence class of (a, b). This is by definition, finding all the elements of A that are equivalente to (a, b).

Let us find all the possible elements of A that are equivalent to (2,4). Let (a, b) R (2,4) Then a+4 = b+2. This implies that a+2 = b. So all the elements of the form (a, a+2) are part of this class.