LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

RELATION & FUNCTION - RELATION AS A SUBSET OF THE CARTESIAN PRODUCT

**Relation As A Subset Of The Cartesian Product –**

**If A & B are any two
non-empty sets, then any subset of A X B is defined as a relation from A &
B **

**For example, suppose A
= {1, 2, 3}, and B = {1, 2, 3, 4}. Then {(2, 3), (2, 4), (1, 3)}, is a relation
in in A X B. Many more relations (subsets) can be selected similarly at random
from our product set A X B.**

**Domain, Co-domain, and
Range of a Relation –**

**Let R be a relation
from set A to set B. Then the set of first entries of the ordered pairs in R is
called the domain.**

**Thus Domain = {a ǀ (a,
b) ∈ R}**

**Similarly the set of second
entries of the ordered pairs in R is called the range.**

**Thus Range = {b ǀ (a, b) ∈ R}**

**The second set B is called the
co-domain of R.**

__For Example.__ – If A = {16, 25, 36,
49}, B = {1, 4, 5, 6} and R be the
relation “is square of ” from A to B then,

** R = {(a, b), ǀ a = b², a ∈ A, b ∈ B}**

** R = {(16, 4), (25, 5), (36, 6)}, Domain of
R = {16, 25, 36}**

**Range of R = {4, 5, 6},
co-domain of R = {1, 4, 5, 6}**

**There are some important note to be remembered which is given below -**

__Remember.1)__ The number of
relations that are possible from a set A of m elements to another set B of n
elements.

**So, number of elements in A = m,
number of elements in B = n,**

**Number of elements in A X B =
mn,**

**So, number of subsets of A X B =
2ᵐⁿ**

**Since every sub-set of A X B is
a relation from A to B, therefore 2ᵐⁿ relations are possible from A to B.**

__Remember.2)__ If R₁ and R₂ are two
relations from A to B, then R₁ ∪ R₂, R₁ ∩ R₂, R₁ -
R₂, are also relations from A to B.

__Remember.3)__ Since ϕ ⊆ A X B,
therefore, ϕ is a relation from A to B. Also Dom ϕ = ϕ and Range ϕ = ϕ.