Relations

Domain and Range of Relation Domain of Relation: The Domain of relation R is the set of elements in P which are related to some elements in Q, or it is the set of all first entries of the ordered pairs in R. It is denoted by DOM (R).

Range of Relation: The range of relation R is the set of elements in Q which are related to some element in P, or it is the set of all second entries of the ordered pairs in R. It is denoted by RAN (R).

Example:

Let A = {1, 2, 3, 4}
B = {a, b, c, d}
R = {(1, a), (1, b), (1, c), (2, b), (2, c), (2, d)}.
Solution:

DOM (R) = {1, 2} RAN (R) = {a, b, c, d}

Complement of a Relation Consider a relation R from a set A to set B. The complement of relation R denoted by R is a relation from A to B such that

R= {(a, b): {a, b) ∉ R}. Example:

Consider the relation R from X to Y
X = {1, 2, 3}
Y = {8, 9}
R = {(1, 8) (2, 8) (1, 9) (3, 9)}
Find the complement relation of R.
Solution:

X x Y = {(1, 8), (2, 8), (3, 8), (1, 9), (2, 9), (3, 9)} Now we find the complement relation R from X x Y R = {(3, 8), (2, 9)}