1. RELATIONS

1.1. What is ‘Relation’?

Let A={ a, b } and B={1, 2, 3 } be two sets. The set of all ordered pairs such that the first element is from the set A and the second element is from the set B is called A cartesian B (cartesian product of A and B) and is denoted:

AxB={ (a,1);(a,2);(a,3);(b,1);(b,2);(b,3) }

Important Remark: AxB ¹ BxA but n(AxB) = n(BxA)

Number of elements of AxB is : n(AxB) = n(A)x n(B)

Number of subsets of AxB is : P(AxB) = 2^n(AxB)

Number of subsets of AxB having ‘r’ elements: P_r(AxB) = C(n,r)............(C(n,r)= n!/ (n-r)!r!)

Definition: A relation from A to B is any subset of AxB and there is 2^n(AxB) subsets from A to B

1.2. Properties of Relations

Let R be a relation defined in E

Reflexive Property: R is said to be reflexive iff "xÎE, ( x, x )ÎR

Symmetric Property: R is said to be symmetric iff " x, y Î E, ( x, y )ÎR Þ ( y, x ) Î R

Anti-symmetric Property: R is said to be anti-symmetric iff " x, y ÎE, ( x, y )Ù( y, x ) Þ x = y

Transitive Property: R is said to be transitive iff " x, y, z ÎE, ( x, y )ÎR Ù ( y, z )ÎR Þ ( x, z )ÎR