Subset
A subset is a set whose every element is also an element of another set, called the superset. If A is a subset of B (written A ⊆ B), then for every element x in A, x is also in B. The concept of subsets is fundamental to set theory and appears throughout mathematics in topics like topology, algebra, and probability theory.
Key Formula
A ⊆ B if and only if every x in A is also in B
LaTeX: A \subseteq B \iff \forall x (x \in A \Rightarrow x \in B)
| Symbol | Meaning | Unit |
|---|---|---|
| A | The potential subset | set |
| B | The potential superset | set |
| x | An arbitrary element | element |
Worked Example
Problem
Let U = {1, 2, 3, 4, 5}, A = {1, 3, 5}, B = {1, 2, 3, 4, 5}. Is A a subset of B? How many subsets does A have?
Solution
Step 1: Check if every element of A is in B. — 1 ∈ B? Yes. 3 ∈ B? Yes. 5 ∈ B? Yes. — Therefore A ⊆ B. Step 2: Count subsets of A = {1, 3, 5} (|A| = 3). — Number of subsets = 2^|A| = 2^3 = 8. — They are: {}, {1}, {3}, {5}, {1,3}, {1,5}, {3,5}, {1,3,5}.
Answer
A ⊆ B (A is a subset of B); A has 8 subsets including the empty set and itself.
Types of Subsets and Their Properties
| Type | Notation | Definition | Example (A={1,2,3}) |
|---|---|---|---|
| Subset | ⊆ | Every element of A is in B (can be equal) | A ⊆ A (true) |
| Proper Subset | ⊂ | A ⊆ B and A ≠ B | {1,2} ⊂ A (true) |
| Empty Set | ∅ ⊆ B | Empty set is subset of every set | ∅ ⊆ A (always true) |
| Power Set | P(A) | Set of all subsets of A | P(A) has 2³=8 elements |
| Superset | ⊇ | B contains all elements of A | A ⊇ {1,2} |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Set Theory
Set theory is the branch of mathematical logic that studies collections of objects, called sets, and the relationships between them. It provides the foundational language for nearly all of modern mathematics, defining concepts like numbers, functions, and relations in terms of sets. Developed formally by Georg Cantor in the 1870s, it underpins areas from algebra and topology to computer science and logic.
Set Union
The union of two sets A and B is the set containing all elements that are in A, in B, or in both, with no element repeated. Written as A ∪ B, it combines the members of both sets into a single collection. Set union is used in probability (combining events), database queries (SQL OR), and logic circuit design.
Set Intersection
The intersection of two sets A and B is the set containing only those elements that are members of both A and B simultaneously. Written as A ∩ B, it identifies common elements shared between collections. Set intersection is used in database queries (SQL JOIN/AND), probability (joint events), and data deduplication tasks.
From Latin "sub" (under/below) + Old English "settan" (to set/place). The mathematical notation ⊆ was introduced in the early 20th century. The concept was central to Cantor's original formulation of set theory.