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Relations & Functions (Basic) revision notes

Every idea in the Relations & Functions (Basic) chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 25 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Cartesian product and relations

Cartesian product and number of relations
10× asked, last 2025
n(A×B)=n(A) n(B)=mn,#relations from A to B=2mnn(A\times B)=n(A)\,n(B)=mn,\qquad \#\text{relations from }A\text{ to }B=2^{mn}
A relation from AA to BB is any subset of A×BA\times B, empty and full included. A relation on AA is a subset of A×AA\times A.
001122334455xxyymax⁡=3\max=3: 7 pairsmax⁡=4\max=4: 9 pairs
max⁡{x,y}=k\max\{x,y\}=k is an L of (k+1)2−k2=2k+1(k+1)^2-k^2=2k+1 cells, so (3,3)(3,3) and (4,4)(4,4) are counted once: 7+9=167+9=16
  • Not adding the counts of independent conditions. If both must hold and share no variable, multiply. A={2,3,4}A=\{2,3,4\}, B={8,9,12}B=\{8,9,12\}, a1∣b2a_1\mid b_2 and a2∣b1a_2\mid b_1: 6 choices each, so 36.
  • Not 4+4+5+54+4+5+5 for max⁡{x,y}∈{3,4}\max\{x,y\}\in\{3,4\} on {0,…,5}\{0,\dots,5\}. That counts (3,3)(3,3) and (4,4)(4,4) twice. Max exactly kk on a set starting at 0: (k+1)2−k2=2k+1(k+1)^2-k^2=2k+1 pairs, so 7+9=167+9=16.
Domain, range and inverse of a relation
3× asked, last 2025
R−1={(b,a):(a,b)∈R},Dom(R−1)=Range(R)R^{-1}=\{(b,a):(a,b)\in R\},\qquad \text{Dom}(R^{-1})=\text{Range}(R)
Domain = first entries used, range = second entries used; the codomain is all of BB. A chain of kk links joins k+1k+1 elements.
xxyyx2+3y2=8x^2+3y^2=8y=1y=1y=−1y=-1
Integer pairs inside: yy is only −1,0,1-1,0,1, so Dom(R−1)=Range(R)={−1,0,1}\text{Dom}(R^{-1})=\text{Range}(R)=\{-1,0,1\}
  • Not the xx-values for the domain of R−1R^{-1}. The inverse swaps each pair, so you need the yy-values. x2+3y2≤8x^2+3y^2\le8 on Z\mathbb Z: y∈{−1,0,1}y\in\{-1,0,1\}, since 3⋅22>83\cdot2^2>8.

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Practise Relations & Functions (Basic)

23 more concepts in 6 topics

Types of relations, Counting relations and closures, Functions and their types, Counting functions, Composition of functions and Greatest integer and fractional part. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Types of relations

  • Reflexive, symmetric, transitive18×
  • Equivalence relations and classes11×
  • Other types of relations2×

Counting relations and closures

  • Counting relations on an n-set2×
  • Relations with forced pairs3×
  • Fewest pairs for reflexive or symmetric8×
  • Smallest equivalence relation containing R5×
  • Counting equivalence relations1×

Functions and their types

  • Function, domain, codomain, range1×
  • Domain of standard expressions3×
  • One-one and many-one11×
  • Onto, into and bijective11×

Counting functions

  • All, one-one and bijective functions6×
  • Onto functions4×
  • Functions with conditions7×

Composition of functions

  • Composition and domain of f∘g8×
  • One-one and onto for composites9×
  • Repeated composition6×
  • Inverse function

Greatest integer and fractional part

  • Greatest integer and fractional part5×
  • Equations and inequalities in [x]9×
  • Range and extreme values with [x]3×
  • Sums with [x]2×

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