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SET and Relation formulas

Master SET and Relation through 38 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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SET and Relation, every formula

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Number of subsets
$n (A) = m \Rightarrow number of subsets = 2^{m}$
m = number of elements in A
Number of proper subsets
$Proper subsets = 2^{m} - 1$
excludes A itself
Power set
$∣ P (A) ∣ = 2^{n (A)}$
P(A) = set of all subsets of A
Union
$A \cup B = {x : x \in A or x \in B}$
Q1MCQUnion
If $A=\{1,2,3\}$ and $B=\{3,4\}$, then $A\cup B$ is:
- A
  $\{1,2,3,4\}$
- B
  $\{3\}$
- C
  $\{1,2\}$
- D
  $\{1,2,3\}$
Intersection
$A \cap B = {x : x \in A and x \in B}$
Q1MCQIntersection
If $A=\{1,2,3,4\}$ and $B=\{2,4,6\}$, then $A\cap B$ is:
- A
  $\{2,4\}$
- B
  $\{6\}$
- C
  $\{1,3\}$
- D
  $\{2,4,6\}$
Difference
$A - B = {x : x \in A, x \in / B} = A \cap B^{c}$
Q1MCQDifference
If $A=\{1,2,3,4\}$ and $B=\{2,4\}$, then $A-B$ is:
- A
  $\{1,3\}$
- B
  $\{2,4\}$
- C
  $\{1,2,3,4\}$
- D
  $\varnothing$
Symmetric difference
$A △ B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)$
Q1MCQSymmetric difference
If $A=\{1,2,3\}$ and $B=\{2,3,4\}$, then $A\,\triangle\,B$ is:
- A
  $\{1,4\}$
- B
  $\{2,3\}$
- C
  $\{1,2,3,4\}$
- D
  $\varnothing$
Complement
$A^{c} = {x \in U : x \in / A}, A \cup A^{c} = U, A \cap A^{c} = \emptyset$
De Morgan's laws
$(A \cup B)^{c} = A^{c} \cap B^{c}, (A \cap B)^{c} = A^{c} \cup B^{c}$
Q1MCQDe Morgan
$(A\cup B)^{c}$ equals:
- A
  $A^{c}\cap B^{c}$
- B
  $A^{c}\cup B^{c}$
- C
  $A\cap B$
- D
  $A^{c}\cap B$
Distributive law
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
and the dual with cup/cap swapped
Q1MCQDistributive
$A\cap(B\cup C)$ equals:
- A
  $(A\cap B)\cup(A\cap C)$
- B
  $(A\cup B)\cap(A\cup C)$
- C
  $(A\cap B)\cap(A\cap C)$
- D
  $A\cup(B\cap C)$
Difference over union
$A - (B \cup C) = (A - B) \cap (A - C)$
Q1MCQDifference over union
$A-(B\cup C)$ equals:
- A
  $(A-B)\cap(A-C)$
- B
  $(A-B)\cup(A-C)$
- C
  $(A\cap B)-C$
- D
  $A-(B\cap C)$
Cardinality of a union
$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
Disjoint union
$A \cap B = \emptyset \Rightarrow n (A \cup B) = n (A) + n (B)$
Cardinality of a difference
$n (A - B) = n (A) - n (A \cap B)$
Neither A nor B
$n (A^{c} \cap B^{c}) = n (U) - n (A \cup B)$
Union of three sets (inclusion-exclusion)
$n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap B) - n (B \cap C) - n (C \cap A) + n (A \cap B \cap C)$
Symmetric difference count
$n (A △ B) = n (A) + n (B) - 2 n (A \cap B)$
Exactly one of A, B, C
$n (A) + n (B) + n (C) - 2 (n (A \cap B) + n (B \cap C) + n (C \cap A)) + 3 n (A \cap B \cap C)$
Exactly two of A, B, C
$n (A \cap B) + n (B \cap C) + n (C \cap A) - 3 n (A \cap B \cap C)$
Bounds on a union
$max (p, q) \leq n (A \cup B) \leq p + q$
p=n(A), q=n(B)
Bounds on an intersection
$0 \leq n (A \cap B) \leq min (p, q)$
p=n(A), q=n(B)
Double complement
$(A^{c})^{c} = A$
Q1MCQDouble complement
$(A^{c})^{c}$ equals:
- A
  $A$
- B
  $A^{c}$
- C
  $U$
- D
  $\varnothing$
Number of relations (JEE)
$Relations from A to B = 2^{n (A) \cdot n (B)}$
a relation is any subset of A x B
Subsets containing a fixed element (JEE)
$2^{n - 1}$
number of subsets of an n-element set that contain a particular element
Cartesian product
$A \times B = {(a, b) : a \in A, b \in B}, n (A \times B) = n (A) n (B)$
Number of relations from A to B
$2^{n (A) \cdot n (B)}$
a relation is any subset of A x B
Number of relations on a set
$Relations on A = 2^{n^{2}}$
n = n(A) (relations from A to A)
Number of reflexive relations
$2^{n^{2} - n}$
n = n(A)
Number of symmetric relations
$2^{n (n + 1) /2}$
n = n(A)
Reflexive and symmetric relations
$2^{n (n - 1) /2}$
n = n(A)
Inverse relation
$R^{- 1} = {(b, a) : (a, b) \in R}, dom R^{- 1} = range R$
Q1MCQInverse relation
If $R=\{(1,2),(3,4)\}$, then $R^{-1}$ is:
- A
  $\{(2,1),(4,3)\}$
- B
  $\{(1,2),(3,4)\}$
- C
  $\{(2,1),(3,4)\}$
- D
  $\{(1,1),(4,4)\}$
Reflexive relation
$(a, a) \in R \forall a \in A$
Q1MCQReflexive relation
A relation $R$ on a set $A$ is reflexive iff:
- A
  $(a,a)\in R\ \forall a\in A$
- B
  $(a,b)\in R\Rightarrow(b,a)\in R$
- C
  $R=\varnothing$
- D
  $(a,b),(b,c)\in R\Rightarrow(a,c)\in R$
Symmetric relation
$(a, b) \in R \Rightarrow (b, a) \in R$
Q1MCQSymmetric relation
If $(2,5)\in R$ and $R$ is symmetric, then $R$ must also contain:
- A
  $(5,2)$
- B
  $(2,2)$
- C
  $(5,5)$
- D
  $(2,5)$ only
Transitive relation
$(a, b) \in R and (b, c) \in R \Rightarrow (a, c) \in R$
Q1MCQTransitive relation
If $(1,2)\in R$, $(2,3)\in R$ and $R$ is transitive, then $R$ must contain:
- A
  $(1,3)$
- B
  $(3,1)$
- C
  $(2,1)$
- D
  $(3,2)$
Equivalence relation
$reflexive + symmetric + transitive$
Q1MCQEquivalence relation
A relation that is reflexive, symmetric and transitive is called:
- A
  an equivalence relation
- B
  a partial order
- C
  a void relation
- D
  an identity relation
Cartesian product over union
$A \times (B \cup C) = (A \times B) \cup (A \times C)$
Q1MCQCartesian over union
$A\times(B\cup C)$ equals:
- A
  $(A\times B)\cup(A\times C)$
- B
  $(A\times B)\cap(A\times C)$
- C
  $(A\cup B)\times C$
- D
  $A\times(B\cap C)$
Intersection of Cartesian products
$(A \times B) \cap (C \times D) = (A \cap C) \times (B \cap D)$
Q1MCQIntersection of products
$(A\times B)\cap(C\times D)$ equals:
- A
  $(A\cap C)\times(B\cap D)$
- B
  $(A\cup C)\times(B\cup D)$
- C
  $(A\cap C)\times(B\cup D)$
- D
  $(A\times C)\cap(B\times D)$
Number of equivalence relations
$= number of partitions of A (Bell number)$

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SET and Relation formulas

SET and Relation, every formula

Number of subsets

Number of proper subsets

Power set

Union

Intersection

Difference

Symmetric difference

Complement

De Morgan's laws

Distributive law

Difference over union

Cardinality of a union

Disjoint union

Cardinality of a difference

Neither A nor B

Union of three sets (inclusion-exclusion)

Symmetric difference count

Exactly one of A, B, C

Exactly two of A, B, C

Bounds on a union

Bounds on an intersection

Double complement

Number of relations (JEE)

Subsets containing a fixed element (JEE)

Cartesian product

Number of relations from A to B

Number of relations on a set

Number of reflexive relations

Number of symmetric relations

Reflexive and symmetric relations

Inverse relation

Reflexive relation

Symmetric relation

Transitive relation

Equivalence relation

Cartesian product over union

Intersection of Cartesian products

Number of equivalence relations

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