RELATION
☞ Cartesian product of sets
Let A and B be non-empty sets. The Cartesian product of A and B denoted by A×B, is the set defined by A×B= {(a,b): aϵA, bϵB}
A×B is the set of all ordered pairs (a,b) the first element of the pair being an element of A and the second being an element of B.
Example1: Let A= {a,b,c}, B={d,e}
A×B= {(a,d), (a,e), (b,d), (b,e), (c,d), (c,e)}
☞ RELATION
Let A and B be two non-empty sets. Then a relation ρ from A to B is a subset of A×B.
Example2: ρ= {(a,d), (c,e)} ⊆ A×B (where A×B is given in example1)
So ρ is a relation from A to B.
☞ Domain of a Relation
For any relation ρ from A to B the domain is the set of all those elements a ϵ A s.t. (a,b) ϵ ρ for some b ϵ B.
Dom(ρ)= {a ϵ A:(a,b) ϵ ρ for some b ϵ B}
From example2 we see Dom(ρ)={a,c}
☞ Range of a Relation
For any relation ρ from A to B the range is the set of all those elements b ϵ B s.t. (a,b) ϵ ρ for some a ϵ A.
Range(ρ)= {b ϵ B:(a,b) ϵ ρ for some a ϵ A}
From EXAMPLE2 we see Range(ρ)={d,e}
☞ Co-domain of a Relation
For any relation ρ from A to B, B is called the co-domain of ρ.
From EXAMPLE2 we see co-domain of ρ= B= {d,e}
☞ Inverse of a Relation
Let ρ be a relation from A to B, the inverse of the relation ρ denoted by ρ-1 is a relation from B to A and defined by ρ-1 = {(b,a):(a,b)ϵρ}
From EXAMPLE2 we see ρ-1= {(d,a), (e,c)}
☞ Relation on a set
Let S be a non-empty set. A relation ρ on S is a subset of S×S.
☞ Types of Relation on a set
⚫Identity Relation
Let S be a non-empty set. A relation ρ on S is said to be an identity relation if ρ= {(a,a):a ϵ S}
Example3: S= {1,2,3}
ρ= {(1,1), (2,2), (3,3)} is an identity relation on S.
⚫Reflexive Relation
Let S be a non-empty set. A relation ρ on S is said to be reflexive if (a,a) ϵ ρ for all a in S.
Example4: S= {1,2,3}
ρ= {(1,1), (2,2), (3,3), (2,3)} is a reflexive relation on S.
⚫Symmetric Relation
A relation ρ on S(non-empty) is said to be symmetric if aρb ⇒ bρa for aϵS ,b ϵ S.
Example5: S= {1,2,3}
ρ= {(2,2), (3,1), (1,3)} is a symmetric relation on S.
⚫Transitive Relation
A relation ρ on S(non-empty) is said to be transitive if for any three elements a,b,c in S, (a,b) ϵ ρ and (b,c) ϵ ρ ⇒(a,c) ϵ ρ.
Example6: S= {1,2,3}
ρ= {(1,1), (2,1), (1,2), (2,2)} is transitive relation on S.
⚫Equivalence Relation
A relation ρ on S(non-empty) is said to be an equivalence relation if ρ is reflexive, symmetric, transitive.
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