# Relations of a Set

MathematicsComputer EngineeringMCA

Relations may exist between objects of the same set or between objects of two or more sets.

## Definition and Properties

A binary relation R from set x to y (written as xRy or R(x,y)) is a subset of the Cartesian product x × y. If the ordered pair of G is reversed, the relation also changes.

Generally an n-ary relation R between sets A1, ... ,\ and\ An is a subset of the n-ary product A1 × ... × An. The minimum cardinality of a relation R is Zero and the maximum is n2 in this case.

A binary relation R on a single set A is a subset of A × A.

For two distinct sets, A and B, having cardinalities m and n respectively, the maximum cardinality of a relation R from A to B is mn.

## Domain and Range

If there are two sets A and B, and relation R have order pair (x, y), then −

• The domain of R, Dom(R), is the set { x | (x, y) ∈ R for some y in B }

• The range of R, Ran(R), is the set { y | (x, y) ∈ R for some x in A}

## Examples

Let, A = { 1, 2, 9 } and B = { 1, 3, 7 }

• Case 1 − If relation R is 'equal to' then R = { (1, 1), (3, 3) }

Dom(R) = { 1, 3 } , Ran(R) = { 1, 3 }

• Case 2 − If relation R is 'less than' then R = { (1, 3), (1, 7), (2, 3), (2, 7) }

Dom(R) = { 1, 2 } , Ran(R) = { 3, 7 }

• Case 3 − If relation R is 'greater than' then R = { (2, 1), (9, 1), (9, 3), (9, 7) }

Dom(R) = { 2, 9 }, Ran(R) = { 1, 3, 7 }