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Home » Math Homework Help » Algebra Homework Help » Relation of Sets
Relation of Sets
A relation R consists of the followings:

(i) Two sets say A and B (ii) an open sentence P(x, y) in which P(a, b) is either true or false for any ordered pair (a, b) A × B, then R is called a relation from A to B and is denoted by

R = {A, B, P(x, y)}.

It may be noted that, here R does not stand for the set of real numbers.

Relation R in a set

If the set B = A, then the relation R is said to be defined in the set A or R is a relation in the set A and

R = {A, A, P(x, y)}.

Further, if P(a, b) is true we say that “a is related to b in the sense of R” and is written as aRb.

In case, P(a, b) is not true we say that a is not related to b in the sense of R and is written as a R b.

Illustration 1: let A = {1, 2, 3} ; B = {1, 6} and P (x, y) = x is less than y, then A × B = {(1, 1), (1, 6), (2, 1), (2, 6), (3, 1), (3, 6)}.

We notice that the following ordered pairs have the property x < y,

(1, 6), (2, 6) and (3, 6).

Hence R = {(1, 6), (2, 6), (3, 6)}.

We also notice that R is a subset of A × B.

Definition of relation:
A relation R from a set A to the set B is a subset of A × B i.e. R ⊆ A × B.

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