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Mathematics

Sets & Set Notation

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Amari Cross & Matthew Williams
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NotationSetsSubsets

Set notation, listing and builder form, universal sets, subsets, and complement.

Sets give mathematics a precise way to talk about collections. The symbols may look small, but they prevent confusion about what belongs, what does not belong, and how groups are related.

CSEC Sets questions often test notation directly, then use that notation in Venn diagrams and word problems. Learn the language first: element, subset, universal set, complement, empty set, and cardinality. Once the notation is clear, the calculations become much easier to follow.

What is a Set?

A set is a well-defined collection of distinct objects called elements or members.

Key characteristics:

  • Elements are distinct (no repeats)
  • A set is well-defined (you must be able to determine whether something belongs or not)
  • Order doesn't matter
Example

Good sets (well-defined):

  • The set of vowels: {a, e, i, o, u}
  • The set of even numbers: {2, 4, 6, 8, ...}
  • The set of months with 31 days

Not sets (not well-defined):

  • "The set of tall people" — what counts as tall?
  • "The set of good books" — good is subjective

Membership: Is Something IN the Set?

If an object is in a set, we say it's a member or element of that set.

Notation:

  • a∈Aa \in A means "a is an element of set A" or "a belongs to A"
  • a∉Aa \notin A means "a is NOT an element of set A"
Example

Let A={1,2,3,5,7}A = \{1, 2, 3, 5, 7\}

  • 3∈A3 \in A ✓ (3 is in the set)
  • 4∉A4 \notin A ✓ (4 is not in the set)
  • 7∈A7 \in A ✓ (7 is in the set)

Cardinality: How Many Elements?

The cardinality of a set is the number of elements in it.

Notation: n(A)n(A) means "the cardinality of set A" or "the number of elements in A"

Example

Let B={a,e,i,o,u}B = \{a, e, i, o, u\}

The cardinality is n(B)=5n(B) = 5 (there are 5 vowels)

Let C={1,2,3,4,5,6,7,8,9,10}C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

The cardinality is n(C)=10n(C) = 10 (there are 10 numbers)

Finite vs. Infinite Sets

  • Finite set: Has a limited number of elements
  • Infinite set: Has unlimited elements
Example

Finite sets:

  • The set of students in your class
  • The set of days in a week: {Mon, Tue, Wed, Thu, Fri, Sat, Sun}
  • The set of planets in our solar system

Infinite sets:

  • The set of natural numbers: {1, 2, 3, 4, ...}
  • The set of integers: {..., -2, -1, 0, 1, 2, ...}
  • The set of all real numbers

Special Sets

Empty Set (or Null Set):

  • Contains NO elements
  • Notation: ∅\emptyset or { }
  • Example: The set of even prime numbers greater than 2

Universal Set:

  • Contains ALL elements being considered in a problem
  • Notation: UU
  • Changes depending on context
Example

If we're discussing "numbers in a classroom game," the universal set might be {1, 2, 3, 4, 5, 6}.

If we're discussing "letters of the alphabet," the universal set is {a, b, c, ..., z}.

In a specific problem, we're told what U is.

Remember
  • Order doesn't matter in sets: {1, 2, 3} = {3, 1, 2}
  • Duplicates aren't allowed: {1, 2, 2, 3} = {1, 2, 3}
  • The empty set ∅ is a valid set
  • Every set is a subset of the universal set U

Part 2: Representing Sets

Sets can be written in three different ways. Choosing the right form makes problems easier!

Method 1: Listing/Roster Form

Write all elements inside curly brackets, separated by commas.

Use this when: The set is small and finite.

Example
  • Set of vowels: A={a,e,i,o,u}A = \{a, e, i, o, u\}
  • Set of digits: B={0,1,2,3,4,5,6,7,8,9}B = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}
  • Set of even numbers less than 10: C={2,4,6,8}C = \{2, 4, 6, 8\}

If a set is infinite but follows a pattern, use "..." to show it continues:

Example
  • Set of natural numbers: N={1,2,3,4,5,...}\mathbb{N} = \{1, 2, 3, 4, 5, ...\}
  • Set of all even numbers: E={2,4,6,8,10,...}E = \{2, 4, 6, 8, 10, ...\}
  • Set of multiples of 5: M={5,10,15,20,...}M = \{5, 10, 15, 20, ...\}

Method 2: Set-Builder Notation

Describe the condition that elements must satisfy.

Format: {x:condition}\{x : \text{condition}\} which reads "the set of all x such that [condition]"

Use this when: It's hard or impossible to list all elements.

Example
  • {x:x is a vowel}\{x : x \text{ is a vowel}\} = the set of all x such that x is a vowel
  • {x:x∈N and x<10}\{x : x \in \mathbb{N} \text{ and } x < 10\} = natural numbers less than 10 = {1, 2, 3, 4, 5, 6, 7, 8, 9}
  • {x:x is even and 5<x<15}\{x : x \text{ is even and } 5 < x < 15\} = {6, 8, 10, 12, 14}
  • {x:x∈Z}\{x : x \in \mathbb{Z}\} = all integers
Tip

Common symbols in set-builder notation:

  • ∈\in means "is an element of"
  • ∉\notin means "is not an element of"
  • <,>,≤,≥<, >, \leq, \geq are inequality signs
  • N\mathbb{N} = natural numbers, Z\mathbb{Z} = integers, Q\mathbb{Q} = rationals, R\mathbb{R} = reals
  • ∧ means "and", ∨ means "or"

Method 3: Descriptive Form

Describe the set in words.

Use this when: Communication matters more than formality.

Example
  • "The set of all even numbers between 1 and 20"
  • "The set of months with exactly 31 days"
  • "The set of factors of 12"
Exam Tip

On CSEC exams, you might need to:

  • Convert listing form to set-builder notation
  • Convert set-builder notation to listing form
  • Identify which form is most appropriate
  • Count elements in a set using any representation

Part 3: Relationships Between Sets

Sets can relate to each other in important ways. Understanding these relationships is crucial.

Subset: Is One Set Inside Another?

Set AA is a subset of set BB if EVERY element of AA is also in BB.

Notation: A⊆BA \subseteq B (read as "A is a subset of B")

Example

Let A={1,3,5}A = \{1, 3, 5\} and B={1,2,3,4,5}B = \{1, 2, 3, 4, 5\}

Is A⊆BA \subseteq B?

  • Is 1 in B? Yes ✓
  • Is 3 in B? Yes ✓
  • Is 5 in B? Yes ✓

All elements of A are in B, so A⊆BA \subseteq B ✓

Example

Let C={2,4,6}C = \{2, 4, 6\} and D={1,2,3,4,5}D = \{1, 2, 3, 4, 5\}

Is C⊆DC \subseteq D?

  • Is 2 in D? Yes ✓
  • Is 4 in D? Yes ✓
  • Is 6 in D? No ✗

Not all elements of C are in D, so C⊈DC \not\subseteq D (C is NOT a subset of D)

Remember

Important subset facts:

  • Every set is a subset of itself: A⊆AA \subseteq A
  • The empty set is a subset of every set: ∅⊆A\emptyset \subseteq A (for any set A)
  • If A⊆BA \subseteq B and B⊆AB \subseteq A, then A=BA = B (the sets are equal)

Equal Sets vs. Equivalent Sets

Equal sets have exactly the same elements.

  • Notation: A=BA = B
  • Example: {1, 2, 3} = {3, 1, 2}

Equivalent sets have the SAME NUMBER of elements (but not necessarily the same elements).

  • Notation: A≡BA \equiv B or "A and B are equivalent"
  • Example: {1, 2, 3} and {a, b, c} are equivalent (both have 3 elements)
Example

Let A={1,2,3}A = \{1, 2, 3\}, B={3,2,1}B = \{3, 2, 1\}, and C={a,b,c}C = \{a, b, c\}

  • Are A and B equal? Yes: A=BA = B (same elements)
  • Are A and C equal? No: A≠CA \neq C (different elements)
  • Are A and C equivalent? Yes: both have cardinality 3

Complement of a Set

The complement of set AA is the set of ALL elements in the universal set UU that are NOT in AA.

Notation: A′A' or AcA^c (read as "A complement" or "not A")

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

Example

Let U={1,2,3,4,5,6,7,8,9,10}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} and A={2,4,6,8,10}A = \{2, 4, 6, 8, 10\} (even numbers)

Then A′={1,3,5,7,9}A' = \{1, 3, 5, 7, 9\} (odd numbers)

This is everything in U that's NOT in A.

Remember
  • A∪A′=UA \cup A' = U (A and its complement together make the universal set)
  • A∩A′=∅A \cap A' = \emptyset (A and its complement have no overlap)
  • (A′)′=A(A')' = A (the complement of a complement is the original set)

Part 4: Subsets and Counting

Finding All Subsets

Every set has multiple subsets. For a set with nn elements, there are exactly 2n2^n subsets (including the empty set and the set itself).

Example

Let A={1,2}A = \{1, 2\}

All subsets of A:

  1. ∅\emptyset (empty set)
  2. {1}\{1\}
  3. {2}\{2\}
  4. {1,2}\{1, 2\} (the set itself)

Total: 22=42^2 = 4 subsets ✓

Notice: Every element can either be "in" or "out" of a subset. That's 2 choices per element, so 2×2=42 \times 2 = 4 total.

Example

Let B={a,b,c}B = \{a, b, c\}

Number of subsets: 23=82^3 = 8

All subsets:

  1. ∅\emptyset
  2. {a}\{a\}
  3. {b}\{b\}
  4. {c}\{c\}
  5. {a,b}\{a, b\}
  6. {a,c}\{a, c\}
  7. {b,c}\{b, c\}
  8. {a,b,c}\{a, b, c\}

That's 8 subsets! ✓

Exam Tip

Finding subsets systematically:

  • 0 elements: just ∅\emptyset (1 subset)
  • 1 element: one subset for each element (3 subsets if 3 elements)
  • 2 elements: all pairs (3 subsets if 3 elements)
  • 3 elements: all triples (1 subset if 3 elements)

For n elements, always: 1+n+(n2)+(n3)+...=2n1 + n + \binom{n}{2} + \binom{n}{3} + ... = 2^n