Sets, Relations and Functions
Board: JEE Main |Exam:JEEMain
JEEMainstudynotesforSets,RelationsandFunctionsbyAjayYadav.ThischaptercoversallconceptsrequiredforJEEMainwithproblem-solvingtechniques.
KeyConcepts
SetsandOperations
Powerset:2ⁿelements.Venndiagrams.n(A∪B)=n(A)+n(B)-n(A∩B).n(A∪B∪C)=n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(C∩A)+n(A∩B∩C).
Relations
Domain,Range.Types:Reflexive,Symmetric,Transitive,Equivalence.NumberofrelationsfromAtoB=2ⁿk(ⁿɛ).
Functions
Domain,Codomain,Range.One-one(injective),Onto(surjective),Bijective.Even/Oddfunctions.Periodicfunctions.Composition,Inverse.
GraphsofFunctions
Identity,Constant,Polynomial,Rational,Modulus,Signum,GreatestInteger,FractionalPart,Exponential,Logarithmic.Transformations:shifting,scaling,reflection.
ImportantFormulas
| Powerset | |P(A)|=2ⁿwheren=|A| |
| n(A∪B) | n(A)+n(B)-n(A∩B) |
| One-one | f(a)=f(b)⇒a=b |
| Onto | Range=Codomain |
SolvedExamples
Example1:Ifn(A)=3,findnumberofrelationsonA.
Solution:2⁵=512.
Example2:Checkiff(x)=x³isone-oneandontoonR.
Solution:Strictlyincreasing⇒one-one.Range=R⇒onto.Bijective.
Example3:Finddomainoff(x)=1/√(x²-4).
Solution:x²-4>0⇒x<-2orx>2.Domain:(-∞,-2)∪(2,∞).
PracticeQuestions
- IfR={(a,b):a-bisdivisibleby5}onZ,showitisequivalence.
- Checkiff(x)=x/(x²+1)isone-one.
- Findrangeoff(x)=x²+2x+3.
- Iff(x)=2x+1,g(x)=x²-1,findfogandgof.
- Classifyfunctions:sinx,|x|, ex, lnx as injective/surjective.