Show that the relation 'R' defined by (a, b) R (c, d) if a+d=b+c is an equivalence relation on set of integers.

An equivalence relation on a set is a binary relation that is reflexive, symmetric, and transitive.

To show that the relation 'R' defined by (a, b) R (c, d) if a+d=b+c is an equivalence relation on set of integers, we need to prove that it is reflexive, symmetric, and transitive.

1. Reflexivity: A relation R is reflexive if every element is related to itself. For all a, b ∈ Z (set of integers), we have (a, b) R (a, b) because a+b = a+b. Hence, R is reflexive.

2. Symmetry: A relation R is symmetric if for every pair of elements that are related, the reverse is also true. For all a, b, c, d ∈ Z, if (a, b) R (c, d) then a+d = b+c. By the commutative property of addition, we can say that b+c = a+d, which implies (c, d) R (a, b). Hence, R is symmetric.

3. Transitivity: A relation R is transitive if whenever an element a is related to an element b, and that element b is related to an element c, then the element a is also related to the element c. For all a, b, c, d, e, f ∈ Z, if (a, b) R (c, d) and (c, d) R (e, f), then a+d = b+c and c+f = d+e. Adding these two equations, we get a+d+c+f = b+c+d+e. Simplifying, we get a+f = b+e, which implies (a, b) R (e, f). Hence, R is transitive.

In conclusion, the relation 'R' defined by (a, b) R (c, d) if a+d=b+c is an equivalence relation on the set of integers because it is reflexive, symmetric, and transitive.

Here is a table summarizing the properties of the relation 'R':

Property	Definition	Proof
Reflexivity	For all a, b ∈ Z, (a, b) R (a, b)	a+b = a+b
Symmetry	For all a, b, c, d ∈ Z, if (a, b) R (c, d) then (c, d) R (a, b)	a+d = b+c implies b+c = a+d
Transitivity	For all a, b, c, d, e, f ∈ Z, if (a, b) R (c, d) and (c, d) R (e, f), then (a, b) R (e, f)	a+d+c+f = b+c+d+e implies a+f = b+e