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)

 

 

 

3 responses

  1. ɬɧŋզզ ʂıɾ

    Liked by 1 person

    1. Ri8

      Like

  2. We should get great help from this
    Thanks sir for your big help on this topic…

    Like

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