Set Theory, including Venn Diagram: formula sheet

Key formulas

Membership

x ∈ A means x is an element of A; x ∉ A means it is not

Use ∈ between an element and a set, never between two sets.

Roster form

A = {a₁, a₂, a₃, ...}

List each element once. Order does not matter.

Set-builder form

A = {x : condition on x}

Read as 'the set of all x such that the condition holds'. Always state which number set x belongs to if it matters.

Natural numbers

N = {1, 2, 3, ...}

Counting numbers. Zero is not included.

Whole numbers

W = {0, 1, 2, 3, ...}

Natural numbers together with zero.

Integers

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

All positive and negative whole numbers and zero.

Rational numbers

Q = {p/q : p, q ∈ Z, q ≠ 0}

Numbers that can be written as a fraction of two integers. Includes terminating and recurring decimals.

Real numbers

R = rational numbers together with irrational numbers

Includes numbers like √2 and π, which are irrational.

Number of subsets

Number of subsets of A = 2ⁿ

n = number of elements in A. Includes ∅ and A itself.

Number of proper subsets

2ⁿ − 1

Excludes only the set A itself. Empty set is counted as a proper subset when A is non-empty.

Number of non-empty proper subsets

2ⁿ − 2

Excludes both ∅ and A.

Number of elements in power set

n[P(A)] = 2ⁿ

Same as the number of subsets.

Subsets of a given size

Number of subsets with r elements = nCr = n! ÷ (r! × (n − r)!)

Use when the question asks for subsets with exactly r elements.

Equal and equivalent sets

A = B if same elements; A ~ B if n(A) = n(B)

Equal implies equivalent, not the other way round.

Subset of an empty set

∅ has exactly 1 subset (itself) and 0 proper subsets

Here n = 0, so 2⁰ = 1 and 2⁰ − 1 = 0.

Union

A ∪ B = {x : x ∈ A or x ∈ B}

"Or" is inclusive. Write common elements only once.

Intersection

A ∩ B = {x : x ∈ A and x ∈ B}

If there is no common element, the answer is ∅.

Difference

A − B = {x : x ∈ A and x ∉ B}

A − B = A ∩ B′. Not the same as B − A in general.

Symmetric difference

A Δ B = (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B)

Elements in exactly one of the two sets.

Complement

A′ = U − A; (A′)′ = A; U′ = ∅; ∅′ = U

Always depends on the universal set U.

Disjoint sets

A ∩ B = ∅, so n(A ∪ B) = n(A) + n(B)

Disjoint means no common element.

Basic properties

A ∪ A = A; A ∩ A = A; A ∪ ∅ = A; A ∩ ∅ = ∅; A ∪ A′ = U; A ∩ A′ = ∅

Union and intersection are commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A.

Subset links

If A ⊆ B then A ∪ B = B, A ∩ B = A and A − B = ∅

Useful for quick checks.

Commutative laws

A ∪ B = B ∪ A; A ∩ B = B ∩ A

Order of the two sets does not matter.

Associative laws

(A ∪ B) ∪ C = A ∪ (B ∪ C); (A ∩ B) ∩ C = A ∩ (B ∩ C)

Works only when the same operation is repeated.

Distributive laws

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C); A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Each operation distributes over the other.

Idempotent laws

A ∪ A = A; A ∩ A = A

A set combined with itself does not change.

Identity laws

A ∪ ∅ = A; A ∩ U = A; A ∪ U = U; A ∩ ∅ = ∅

∅ is the identity for union, U is the identity for intersection.

Complement laws

A ∪ A′ = U; A ∩ A′ = ∅; (A′)′ = A; ∅′ = U; U′ = ∅

A set and its complement together make up U and share nothing.

De Morgan's laws

(A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′

Flip ∪ to ∩ (or the reverse) and complement each set.

Difference in terms of complement

A − B = A ∩ B′

Useful for converting differences before applying laws.

Union of two sets

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Subtract the overlap once because it is counted twice when you add n(A) and n(B).

Difference of sets

n(A − B) = n(A) − n(A ∩ B)

A − B is only A, the part of A outside B. It is not the same as B − A.

Complement

n(A′) = n(U) − n(A)

A′ is everything in the universal set that is not in A.

Union of three sets

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C)

Add the singles, subtract the pairs, add back the triple overlap.

Neither set

n(neither A nor B) = n(U) − n(A ∪ B)

This is the region outside both circles, which is (A ∪ B)′.

Two sets (union)

n(A∪B) = n(A) + n(B) − n(A∩B)

Valid for finite sets. If A and B are disjoint, n(A∩B) = 0.

Three sets (union)

n(A∪B∪C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)

Add singles, subtract pairs, add the triple.

Only A (two sets)

n(A only) = n(A) − n(A∩B)

Same as n(A − B).

Only A (three sets)

n(only A) = n(A) − n(A∩B) − n(A∩C) + n(A∩B∩C)

Remove both overlaps but add back the triple, which was removed twice.

Neither / none

n(none) = n(U) − n(A∪B∪C)

Complement of the union. Written n(A′∩B′∩C′).

Exactly two of three

Exactly two = n(A∩B) + n(B∩C) + n(A∩C) − 3·n(A∩B∩C)

Each pair overlap includes the triple region, so subtract it three times.

Cartesian product

A × B = {(a, b) : a ∈ A, b ∈ B}

Order matters. In general A × B ≠ B × A.

Number of elements in A × B

n(A × B) = n(A) × n(B)

Holds for finite sets. If either set is empty, the product is empty.

Number of relations from A to B

2^(m × n), where m = n(A) and n = n(B)

A relation is a subset of A × B, and a set with mn elements has 2^(mn) subsets. This count includes the empty relation and A × B itself.

Domain and range of a relation R

Domain = {a : (a, b) ∈ R}; Range = {b : (a, b) ∈ R}

Range is a subset of the codomain B. It need not equal B.

Equality of ordered pairs

(a, b) = (c, d) if and only if a = c and b = d

Use this to find unknown values from given equal pairs.

Product with itself

n(A × A) = [n(A)]²

The number of relations on A is then 2^([n(A)]²).

Quick revision

• A set is a well-defined collection; order and repetition do not matter.
• A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets.
• The empty set φ is a subset of every set.
• A − B contains elements in A but not in B.
• A′ = U − A, the elements of U that are not in A.
• De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
• n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
• For disjoint sets, n(A ∩ B) = 0, so n(A ∪ B) = n(A) + n(B).
• Only A only = n(A) − n(A ∩ B).
• Neither A nor B = n(U) − n(A ∪ B).
• n(A × B) = n(A) × n(B), and A × B is generally not equal to B × A.
• (a, b) = (c, d) only when a = c and b = d.

Common mistakes

• Counting repeated elements twice, for example saying the set of letters in 'MISSISSIPPI' has 11 elements. Fix: List only distinct letters: {M, I, S, P}. That gives 4 elements.
• Including 0 in the natural numbers. Fix: Remember N starts at 1 and W starts at 0.
• Counting repeated elements when finding n Fix: Rewrite the set with distinct elements first. {1, 2, 2, 3} has n = 3.
• Treating {0} or {∅} as the empty set Fix: Both contain one element, so each is a singleton with n = 1.
• Treating A − B and B − A as the same set. Fix: Always ask "in the first, not in the second". Check which set comes first.
• Writing common elements twice in a union. Fix: After joining, scan and keep each element once. For example {1, 2, 3} ∪ {3, 4} = {1, 2, 3, 4}.
• Writing (A ∪ B)′ = A′ ∪ B′ Fix: Say it aloud: complement of union is intersection of complements. The sign always flips.
• Treating the distributive law as true in only one direction Fix: For sets, both A ∪ (B ∩ C) and A ∩ (B ∪ C) distribute. Learn both forms.
• Treating 'n(A ∩ B) = 12' as the number of elements only in both and then subtracting it again from the overlap region. Fix: In a three-set diagram, the pair total includes the centre. Subtract n(A ∩ B ∩ C) to get the part of the pair region without C.
• Writing the full set total in the 'only A' region. Fix: n(A) is the whole circle, with all overlaps. Subtract the overlaps before writing the 'only A' number.

Exam tips

• Convert any set-builder form to roster form first. Most MCQs become simple matching after that.
• Check the boundary elements of the options. The wrong options usually differ there.
• Memorise the symbols N, W, Z, Q, R and what each contains. Direct questions on them are easy marks.
• For 'well-defined' questions, reject any collection that depends on opinion.
• There is no negative marking, so attempt every question and eliminate options before guessing.
• Memorise powers of 2 up to 2¹⁰ = 1024. Most subset questions are solved by recall.
• Read the wording carefully: subsets, proper subsets and non-empty proper subsets give three different answers.
• Watch for traps like {0}, {∅} and sets with repeated elements before counting.