Math · Sets, Relations and Functions

Sets formulas for JEE

Every Sets formula you need for JEE, grouped by concept.

FormulasRevision notes
34 formulas3 concepts
01Subsets and Power Sets02Set Operations03Properties of Complement Sets
01

Subsets and Power Sets

4 formulas

Set Equality Condition

A⊂B and B⊂A  ⟺  A=BA \subset B \text{ and } B \subset A \iff A = B

Two sets are equal if and only if they are subsets of each other.

setsequalitysubsets

Length of an Interval

L=b−aL = b - a

The length of any interval bounded by a and b on the real number line.

applies whena<ba < b
setsintervalslength

Power set size

∣P(A)∣=2∣A∣|P(A)|=2^{|A|}

The number of subsets in a power set for a finite set A.

applies whenSet A must be finite.
setspowersetcardinalityjee

Subset Condition

a∈A  ⟹  a∈Ba \in A \implies a \in B

Definition of a subset; every element of A is also in B.

applies whenA is a subset of B (A⊂BA \subset B).
setssubsetsdefinition
02

Set Operations

21 formulas

Absorption Law (Intersection)

A∩(A∪B)=AA \cap (A \cup B) = A

The intersection of a set with its union with another set is the set itself.

setsabsorptionintersectionidentities

Absorption Law (Union)

A∪(A∩B)=AA \cup (A \cap B) = A

The union of a set with its intersection with another set is the set itself.

setsabsorptionunionidentities

Set Decomposition

A=(A∩B)∪(A−B)A = (A \cap B) \cup (A - B)

A set A can be partitioned into its intersection with B and its difference from B.

setsdecompositiondifference

Difference of Sets

A−B={x:x∈A and x∉B}A - B = \{x : x \in A \text{ and } x \notin B\}

Set of elements which belong to A but not to B.

setsoperationsdifference

Distributive Law

A∩(B∪C)=(A∩B)∪(A∩C)A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

Intersection distributes over union.

setsintersectionuniondistributive

Associative Law (Intersection)

(A∩B)∩C=A∩(B∩C)(A \cap B) \cap C = A \cap (B \cap C)

The grouping of sets does not matter for intersection.

setsintersectionassociative

Commutative Law (Intersection)

A∩B=B∩AA \cap B = B \cap A

The order of sets does not matter for intersection.

setsintersectioncommutative

Idempotent Law (Intersection)

A∩A=AA \cap A = A

The intersection of a set with itself is the set itself.

setsintersectionidempotent

Law of U (Intersection)

U∩A=AU \cap A = A

The intersection of any set with the universal set is the set itself.

setsintersectionidentity

Intersection of Sets

A∩B={x:x∈A and x∈B}A \cap B = \{x : x \in A \text{ and } x \in B\}

Set of all elements which are common to both A and B.

setsoperationsintersection

Law of Empty Set (Intersection)

ϕ∩A=ϕ\phi \cap A = \phi

The intersection of any set with the empty set is the empty set.

setsintersectionempty_set

Symmetric Difference

AΔB=(A−B)∪(B−A)A \Delta B = (A - B) \cup (B - A)

Set of elements that belong to exactly one of the sets A or B.

setssymmetric_differencejee

Associative Law (Union)

(A∪B)∪C=A∪(B∪C)(A \cup B) \cup C = A \cup (B \cup C)

The grouping of sets does not matter for union.

setsunionassociative

Commutative Law (Union)

A∪B=B∪AA \cup B = B \cup A

The order of sets does not matter for union.

setsunioncommutative

Union of Sets

A∪B={x:x∈A or x∈B}A \cup B = \{x : x \in A \text{ or } x \in B\}

Set consisting of all elements that are either in A, in B, or in both.

setsoperationsunion

Union with Difference Identity

A∪(B−A)=A∪BA \cup (B - A) = A \cup B

The union of A and the elements strictly in B but not A equals the union of A and B.

setsuniondifferenceidentities

Idempotent Law (Union)

A∪A=AA \cup A = A

The union of a set with itself is the set itself.

setsunionidempotent

Law of Identity Element (Union)

A∪ϕ=AA \cup \phi = A

The union of any set with the empty set is the set itself.

setsunionidentity

Union size

∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A\cup B|=|A|+|B|-|A\cap B|

Principle of inclusion-exclusion for the cardinality of the union of two sets.

applies whenSets A and B must be finite.
setscardinalityinclusion_exclusionjee

Union Size (3 Sets)

∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣B∩C∣−∣A∩C∣+∣A∩B∩C∣|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|

Principle of inclusion-exclusion for the cardinality of three intersecting finite sets.

applies whenSets A, B, and C must be finite.
setscardinalityinclusion_exclusionjee

Law of U (Union)

U∪A=UU \cup A = U

The union of the universal set with any of its subsets is the universal set.

setsunionuniversal
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03

Properties of Complement Sets

9 formulas

Complement size

∣Ac∣=∣U∣−∣A∣|A^c|=|U|-|A|

Cardinality of complement in universal set.

applies whenFinite universal set U.
setcardinality

Complement of a Set

A′={x:x∈U and x∉A}A' = \{x : x \in U \text{ and } x \notin A\}

Set of all elements of the universal set which are not elements of A.

setscomplementdefinition

Complement Difference Relation

A′=U−AA' = U - A

The complement of A is the difference between the universal set and A.

setscomplementdifference

Complement Law (Intersection)

A∩A′=ϕA \cap A' = \phi

The intersection of a set and its complement is the empty set.

setscomplementintersection

Complement Law (Union)

A∪A′=UA \cup A' = U

The union of a set and its complement is the universal set.

setscomplementunion

De Morgan's Laws

(A∪B)′=A′∩B′, (A∩B)′=A′∪B′(A \cup B)' = A' \cap B', \ (A \cap B)' = A' \cup B'

The complement of the union/intersection of two sets is the intersection/union of their complements.

setsdemorgancomplement

Law of Double Complementation

(A′)′=A(A')' = A

The complement of the complement of a set is the set itself.

setscomplementdouble

Law of Empty Set Complement

ϕ′=U\phi' = U

The complement of the empty set is the universal set.

setscomplementempty_set

Law of Universal Set Complement

U′=ϕU' = \phi

The complement of the universal set is the empty set.

setscomplementuniversal_set
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