SET and Relation formulas

Every formula JEE Advanced expects you to know for SET and Relation, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 38
Subject: Mathematics
Exam: JEE Advanced
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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
$A \cup B = {x : x \in A or x \in B}$
Intersection
$A \cap B = {x : x \in A and x \in B}$
Difference
$A - B = {x : x \in A, x \in / B} = A \cap B^{c}$
Symmetric difference
$A △ B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)$
Complement
$A^{c} = {x \in U : x \in / A}, A \cup A^{c} = U, A \cap A^{c} = \emptyset$
De Morgan's laws
$(A \cup B)^{c} = A^{c} \cap B^{c}, (A \cap B)^{c} = A^{c} \cup B^{c}$
Distributive law
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
and the dual with cup/cap swapped
Difference over union
$A - (B \cup C) = (A - B) \cap (A - C)$
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
$(A^{c})^{c} = A$
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
$R^{- 1} = {(b, a) : (a, b) \in R}, dom R^{- 1} = range R$
Reflexive relation
$(a, a) \in R \forall a \in A$
Symmetric relation
$(a, b) \in R \Rightarrow (b, a) \in R$
Transitive relation
$(a, b) \in R and (b, c) \in R \Rightarrow (a, c) \in R$
Equivalence relation
$reflexive + symmetric + transitive$
Cartesian product over union
$A \times (B \cup C) = (A \times B) \cup (A \times C)$
Intersection of Cartesian products
$(A \times B) \cap (C \times D) = (A \cap C) \times (B \cap D)$
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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