Relations and Functions

Board: CBSE |Class:Class12

ComprehensivestudynotesforRelationsandFunctionsbyAjayYadav.

KeyConcepts

TypesofRelations

Emptyrelation:R=φ.Universal:R=A×A.Reflexive:(a,a)&inR∀a.Symmetric:(a,b)&inR⇒(b,a)&inR.Transitive:(a,b),(b,c)⇒(a,c).Equivalencerelation=reflexive+symmetric+transitive.

EquivalenceClasses

Thesetofallelementsrelatedtoagivenelementaformsanequivalenceclass[a].Equivalenceclassespartitionthesetintodisjointsubsets.

Functions

f:A→B.One-one(injective):f(a)=f(b)⇒a=b.Onto(surjective):range=codomain.Bijective:bothone-oneandonto.

CompositionofFunctions

(fog)(x)=f(g(x)).Domain:x&indom(g)suchthatg(x)&indom(f).Compositionisassociative:(fog)oh=fo(goh).

InvertibleFunctions

fisinvertibleiffitisbijective.Inversef⁻¹(y)=xifff(x)=y.fof⁻¹(x)=f⁻¹of(x)=x.

ImportantFormulas

Equivalence	Reflexive+Symmetric+Transitive
Composition	(fog)(x)=f(g(x))
Inversecondition	fisinvertibleifffisbijective
fof⁻¹	f(f⁻¹(x))=x=f⁻¹(f(x))

SolvedExamples

Example1:CheckifR={(1,1),(2,2),(3,3),(1,2),(2,1)}on{1,2,3}isequivalence.

Solution:Reflexive:yes.Symmetric:yes.Transitive:needcheck(1,2),(2,1)→(1,1)OK.Yes,equivalence.

Example2:Iff(x)=x²andg(x)=2x+1,findfogandgof.

Solution:fog(x)=f(g(x))=(2x+1)².gof(x)=g(f(x))=2x²+1.

Example3:Checkiff(x)=3x+2isinvertibleonR.

Solution:fislinear→one-one.Range=R→onto.f⁻¹(y)=(y-2)/3.Yes.

PracticeQuestions

1. ShowR={(a,b):|a-b| is even} on Z is equivalence.
2. Find fog if f(x)=√x, g(x)=x²+1.
3. Check if f(x)=x³ is bijective from R to R.
4. If f(x)=2x+3, find f⁻¹(x).
5. Give example of a relation that is symmetric but not transitive.

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