What is the difference between $\subset$ and $\subseteq$?

The symbol $\subseteq$ means "is a subset of or equal to" — every element of $A$ is also in $B$, and $A$ may equal $B$. The symbol $\subset$ is often used for a proper subset, meaning $A \subseteq B$ and $A \neq B$ (so $B$ has at least one element not in $A$). Some textbooks use $\subset$ to mean $\subseteq$, so always check the convention your course follows.

Sets and Set Notation — Definition, Formula & Examples

A set is a well-defined collection of distinct objects, called elements. Set notation is the system of symbols and conventions used to describe sets, their elements, and the relationships between them.

A set $S$ is an unordered collection of distinct elements. A set may be specified by roster notation, which explicitly lists its elements within braces (e.g., $\{1, 2, 3\}$ ), or by set-builder notation, which defines elements by a property they satisfy (e.g., $\{x \mid x \in \mathbb{Z},\; x > 0\}$ ). Two sets $A$ and $B$ are equal if and only if they contain exactly the same elements.

Key Formula

A = \{x \mid P(x)\}

Where:

$A$ = The set being defined
$x$ = A variable representing a potential element
$P(x)$ = A condition (predicate) that each element must satisfy

How It Works

You describe a set in one of two main ways. Roster notation lists every element inside curly braces:

A = \{2, 4, 6, 8\}

. Set-builder notation states a rule:

A = \{x \mid x \text{ is an even integer and } 1 < x < 9\}

. Key symbols include

\in

("is an element of"),

\notin

("is not an element of"),

\subseteq

("is a subset of"),

\cup

(union),

\cap

(intersection), and

\emptyset

(the empty set). Because sets are unordered and contain no duplicates,

\{3, 1, 2\} = \{1, 2, 3\}

and

\{1, 1, 2\} = \{1, 2\}

Worked Example

Problem: Let

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

A = \{2, 4, 6, 8\}

, and

B = \{1, 2, 3, 4, 5\}

. Find

A \cap B

A \cup B

, and determine whether

5 \in A

Step 1: Find the intersection: The intersection

A \cap B

contains elements that are in both

A

and

B

A \cap B = \{2, 4\}

Step 2: Find the union: The union

A \cup B

contains all elements that are in

A

B

(or both), with no duplicates.

A \cup B = \{1, 2, 3, 4, 5, 6, 8\}

Step 3: Test membership: Check whether 5 appears in

A = \{2, 4, 6, 8\}

. It does not.

5 \notin A

Answer:

A \cap B = \{2, 4\}

A \cup B = \{1, 2, 3, 4, 5, 6, 8\}

, and

5 \notin A

Another Example

Problem: Write the set of all positive multiples of 3 that are less than 20 in both roster notation and set-builder notation.

Step 1: List the elements (roster notation): Start at 3 and add 3 repeatedly, stopping before 20.

M = \{3, 6, 9, 12, 15, 18\}

Step 2: Write the rule (set-builder notation): Describe the pattern using a condition on the variable.

M = \{x \in \mathbb{Z}^{+} \mid x \text{ is a multiple of } 3 \text{ and } x < 20\}

Answer: Roster:

\{3, 6, 9, 12, 15, 18\}

. Set-builder:

\{x \in \mathbb{Z}^{+} \mid 3 \mid x \text{ and } x < 20\}

Why It Matters

Set notation is foundational in Algebra 2, Precalculus, and any college course involving discrete mathematics or proofs. Probability theory defines events as sets of outcomes, so fluency with unions, intersections, and complements directly applies to calculating probabilities. Database queries, Venn diagram reasoning, and formal logic all rely on the language of sets.

Common Mistakes

Mistake: Confusing the symbols

\in

and

\subseteq

. For example, writing

\{2\} \in \{1, 2, 3\}

Correction:

\in

relates an element to a set:

2 \in \{1, 2, 3\}

\subseteq

relates a set to a set:

\{2\} \subseteq \{1, 2, 3\}

. The object

\{2\}

is a set, not an element of

\{1, 2, 3\}

(unless the set explicitly contains

\{2\}

as an element).

Mistake: Treating sets as ordered or allowing duplicates, such as thinking

\{1, 2, 3\} \neq \{3, 2, 1\}

Correction: Sets are unordered and contain no repeated elements.

\{1, 2, 3\}

and

\{3, 2, 1\}

are the same set. If order matters, use an ordered tuple like

(1, 2, 3)

Related Terms

Element of a Set — Individual object contained within a set
Empty Set — The unique set with no elements
Intersection — Elements common to two or more sets
Complement of a Set — Elements in the universal set not in a given set
Cardinality — The number of elements in a set
Partition of a Set — Dividing a set into non-overlapping subsets
Finite — A set with a countable, limited number of elements
Infinite — A set with unlimited elements