Higher Secondary Study Material
Plus 1 (Class 11) Mathematics
Chapter 1
Relations and Functions
Let A = {1,2,3} and B = {4,5}, write A×B and B×A?
A×B = {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}
B×A = {(4,1),(4,2),(4,3),(5,1),(5,2),(5,3)}
n(A×B) = n(A).n(B)
Relation
1. Consider two non empty sets A and B. A relation from A to B is always a subset of A×B.
2. A×B is a relation called universal relation and null set is a relation called empty relation
3. Relations are usually denoted by R
4. The set of first elements of all ordered pairs in R is called domain and set of second elements of all ordered pairs in R is called range
5. The number of relations from A to B is 2mn where m is number of elements in A and n is number of elements in B
6. (a,b) ∈ R means a is related to b
Eg:
Consider A = {1,2,3} and B = {4,5,6}. R is a relation from A to B given by
R = {(1,4), (2,5), (3,6)}
A × B = {(1,4),(1,5),(1,6), (2,4),(2,5),(2,6),(3,4),(3,5),(3,6)}
Clearly R is a subset of A×B
Domain = {1,2,3}, Range = {4,5,6}
Number of relations from A to B = 2mn
= 23x3 = 29
Types of Relations
1. Reflexive relation :
A relation R in the set A is called reflexive if every element in A is related to itself. (a,a) ∈ R for every a∈A
2. symmetric Relation :
A relation R in the set A is called symmetric, if a is related to b then b is related to a for all a,b in A
(a,b) ∈ R ⇒ (b,a) ∈ R for all a,b ∈ A
3. Transitive relation :
A relation R in the set A is called transitive if a is related to b and b is related to c then a is related to c for all a,b,c in A
(a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R for all a,b,c ∈ A
4. Equivalence Relation :
A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive
Eg: 1
Let A = {1,2,3}.consider the following relations on A
R1 = {(1,1), (2,2),(2,3),(3,3)}
R1 is a reflexive relation since (a,a) ∈ R for every a∈A
R2 = {(1,1), (2,3), (3,3), (1,3)}
R2 is not reflexive since (2,2)∉ R2
Eg: 2
Let A = {1,2,3}.consider the following relations on A
R1 = {(1,2),(2,1),(1,1), (3,2), (2,3)}
R1 is a symmetric relation since (a,b) ∈ R1 ⇒ (b,a) ∈ R1 for all a,b ∈ A
R2 = {(1,2), (2,1), (3,2)}
R2 is not symmetric since (3,2) ∈ R2 but (2,3) ∉ R2
Eg : 3
Let A = {1,2,3}.consider the following relations on A
R1 = {(1,2),(2,3),(1,3)}
R1 is a transitive relation since (a,b) ∈ R1 and (b,c) ∈ R1 ⇒ (a,c) ∈ R1
R2 = {(1,2),(2,3),(1,3),(3,2),(2,1)}
R2 is not transitive since (3,2) ∈ R2 and (2,1) ∈ R2 but (3,1) ∉ R2 
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