Binary RelationLet P and Q be two non empty sets. A binary relation R is defined to be a subset of P x Q from a set P to Q. If (a, b) ∈ R and R ⊆ P x Q then a is related to b by R i.e., aRb. If sets P and Q are equal, then we say R ⊆ P x P is a relation on P e.g. Example1: If a set has n elements, how many relations are there from A to A. Solution: If a set A has n elements, A x A has n^{2} elements. So, there are 2^{n2} relations from A to A. Example2: If A has m elements and B has n elements. How many relations are there from A to B and vice versa? Solution: There are m x n elements; hence there are 2^{m x n} relations from A to A. Example3: If a set A = {1, 2}. Determine all relations from A to A. Solution: There are 2^{2}= 4 elements i.e., {(1, 2), (2, 1), (1, 1), (2, 2)} in A x A. So, there are 2^{4}= 16 relations from A to A. i.e. Domain and Range of RelationDomain 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: Solution: DOM (R) = {1, 2} RAN (R) = {a, b, c, d} Complement of a RelationConsider 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: 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)}
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