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Sets formula sheet

Every formula in the Sets chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 13 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 formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Sets, types and subsets

Roster and set-builder form
7× asked, last 2025
{x∈Z:x2<9}={−2,−1,0,1,2}\{x\in\mathbb Z: x^2<9\}=\{-2,-1,0,1,2\}
Each element once; order and repeats do not matter. N={1,2,3,… }\mathbb N=\{1,2,3,\dots\}. Count distinct values (2/4=1/22/4=1/2).
Types of sets
n(∅)=0,n({∅})=1n(\varnothing)=0,\qquad n(\{\varnothing\})=1
Equal: same elements. Equivalent: same number of elements.
Subsets and power set
4× asked, last 2024
n(P(A))=2n,proper=2n−1non-empty proper=2n−2\begin{aligned}&n(P(A))=2^n,\qquad \text{proper}=2^n-1\\&\text{non-empty proper}=2^n-2\end{aligned}
n=n(A)n=n(A). n(P(P(A)))=22nn(P(P(A)))=2^{2^n}. 2m−2n=k2^m-2^n=k: the power of 2 in kk gives nn.
Subsets with a condition
5× asked, last 2026
#{C⊆A:C∩B≠∅}=2n(A)−2n(A−B)#{C⊆A:x∈C}=2n(A)−1\begin{aligned}&\#\{C\subseteq A: C\cap B\ne\varnothing\}=2^{n(A)}-2^{n(A-B)}\\&\#\{C\subseteq A: x\in C\}=2^{n(A)-1}\end{aligned}
Count the complement. kk elements forced in: 2n−k2^{n-k}. "p∉Tp\notin T or q∈Tq\in T": 2n−2n−22^n-2^{n-2}.

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9 more formulas in 3 topics

Intervals and set operations, Cardinality and Venn diagrams and Cartesian product and point sets. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Intervals and set operations

  • Intervals10×
  • Operations6×
  • De Morgan's laws3×

Cardinality and Venn diagrams

  • Two sets7×
  • Three sets4×
  • Exactly one, exactly two3×
  • Largest and smallest overlap4×

Cartesian product and point sets

  • Cartesian product1×
  • Point sets3×

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About 40 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

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