Relations- Domain, Range

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Consider the following sentences:Consider the following sentences:

42 is a multiple of 6.

Line l is parallel to line m.

10 is greater than 7.

In each case the first element is related to the last element by the relation in italics. The two elements named are members of two separate sets, say set A and set B.

If and we obtain a set of ordered pairs (x, y). Thus a relation forms a set of ordered pairs.

Relation

If A and B are two non-empty sets, then a relation R in A x B is a subset of A x B. If we use the letter R to denote a relation, then we can write the relation between a and b as aRb a is related to b.

Let R mean 'is greater than', then xRy means x > y.

If A = {1, 2, 3} and B = {3, 5} list the ordered pairs for each of the following relations:

(i)

(ii)

(iii)

A x B = {(1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5)}

(i)

= {(3, 3)} is the only pair which satisfies the relation x = y.

R 'is equal to' is a subset of A x B.

(ii) S = {(x, y) : x, y A x B, y = x + 2}

= {(1, 3), (3, 5)}. These two pairs satisfy the relation y = x + 2.

(iii)

= {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5)}

• Five pairs satisfy the relation x < y

• R, S and T are all subsets of A x B, hence they are all relations.

Domain

The set of first elements in a relation is called Domain.

In example (6), Domain is set A = {1, 2, 3}.

Range

The set of second elements in a relation is called Range.

In example (6), Range is set B = {3, 5}.