# Relations & Functions

## Cartesian Product—

♦An ordered pair⟹(x,y)

♦An ordered triplet⟹(x,y,z)

♦Equality of ordered pairs ⟹ (x,y)=(α,β)⟺x=α,y=β

♦Equality of ordered triplets ⟹ (x,y,z)=(α,β,γ)⟺x=α, y=β, z=γ

Key Results: I.A  (B C)=(A B)⋃(A C)

II.A (B⋂C)=(A B)⋂(A C)

III. A (B-C)=(A B)-(A C)

♦If A and B are two non-empty sets, then A B=B A⟺A=B

If A⊆B; then A A (A B)⋂(B A)

♦A⊆B⟹A C ⊆B C for any set C

♦A⊆B and C⊆D⟹A C ⊆ B D

♦(A B)⋃(C D)⊆(A C) (B⋃D)

♦(A B)⋂(C D)=(A⋂C) (B⋂D)

♦(A B)⋂(B A)=(A⋂B) (B⋂A)

♦Let A and B be two non-empty sets having n elements in common, then A B and B A  have n2 elements in common.

## Relations—

If A and B are any two non empty sets, then each subset of AxB is called a relation from A to B. It’s denoted by R:A⟶B

Key Point : I. If n(A)=p & n(B)=q, then the total number of relations from A to B =2pq

Types of relations:

I.Empty Relation: A relation having no element is called an empty relation.

Ex: R = {x : x+5=0 where x∊N}={}=ϕ

II.Universal Relation: If R:A⟶B is any relation such that R=AxB, then it’s called a universal relation.

Ex: A={1,2} & B={4,5}⟹AxB={(1,4),(1,5),(2,4),(2,5)}

R={(x,y):x+y<10 where x∊A, y∊B} = AxB

III. Inverse Relation: If R:A⟶B  is any relation then its inverse relation R-1:B⟶A is defined as follows:  R-1 = {(y,x):(x,y)∊ R}

Ex: R={(1,2),(1,3),(2,2),(2,3)} ⟹ R-1={(2,1),(3,1),(2,2),(3,2)}

## Functions—

 Name of the Functions Format Domain Range Constant Function y=ƒ(x)=k, k∊R R {k} Identity Function⟹I y=ƒ(x)=x R R Square Function y=ƒ(x)=x2 R [0,∞) Cube Function y=ƒ(x)=x3 R R Power Function y=ƒ(x)=xn R R:when n is odd [0,∞): when n is even Linear Function y=ƒ(x)=ax+b, a≠0 R R Quadratic Function y=ƒ(x)=ax2+bx+c, a≠0 R Calculate Polynomial Function y=ƒ(x)=a0xn+a1xn-1+…an, where n,(n-1),…∊W & a0,a1..an∊R R Calculate Rational Function y=ƒ(x)= ,h(x)≠0 R-{x:h(x)=0} Calculate Irrational Function Having fractional powers of x Calculate Calculate Square Root Function y=ƒ(x)= [0,∞) [0,∞) Cube Root Function y=ƒ(x)= R R Signum Function y=ƒ(x)= R {-1,0,1} Modulus Function y=|x| R [0,∞) Exponential Function y=ax ,a>0,a≠1 R (0,∞) Logarithmic Function y=logax, a>0,a≠1,x>0 (0,∞) R Greatest Integer Function y=[x] R I Least Integer Function y= R I Fractional Part Function y={x} R [0,1)