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---
category: Algorithms & Data Structures
name: Set theory
contributors:
---
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Set theory is a branch of mathematics that studies sets, their operations, and their properties.
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* A set is a collection of disjoint items.
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## Basic symbols
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### Operators
* the union operator, `∪ ` , pronounced "cup", means "or";
* the intersection operator, `∩` , pronounced "cap", means "and";
* the exclusion operator, `\` , means "without";
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* the complement operator, `'` , means "the inverse of";
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* the cross operator, `× ` , means "the Cartesian product of".
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### Qualifiers
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* the colon, `:` , or the vertical bar `|` qualifiers are interchangeable and mean "such that";
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* the membership qualifier, `∈` , means "belongs to";
* the subset qualifier, `⊆` , means "is a subset of";
* the proper subset qualifier, `⊂` , means "is a subset of but is not equal to".
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### Canonical sets
* `∅` , the empty set, i.e. the set containing no items;
* `ℕ ` , the set of all natural numbers;
* `ℤ ` , the set of all integers;
* `ℚ ` , the set of all rational numbers;
* `ℝ ` , the set of all real numbers.
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There are a few caveats to mention regarding the canonical sets:
1. Even though the empty set contains no items, the empty set is a subset of itself (and indeed every other set);
2. Mathematicians generally do not universally agree on whether zero is a natural number, and textbooks will typically explicitly state whether or not the author considers zero to be a natural number.
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### Cardinality
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The cardinality, or size, of a set is determined by the number of items in the set. The cardinality operator is given by a double pipe, `|...|` .
For example, if `S = { 1, 2, 4 }` , then `|S| = 3` .
### The Empty Set
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* The empty set can be constructed in set builder notation using impossible conditions, e.g. `∅ = { x : x ≠ x }` , or `∅ = { x : x ∈ N, x < 0 }` ;
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* the empty set is always unique (i.e. there is one and only one empty set);
* the empty set is a subset of all sets;
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* the cardinality of the empty set is 0, i.e. `|∅| = 0` .
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## Representing sets
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### Literal Sets
A set can be constructed literally by supplying a complete list of objects contained in the set. For example, `S = { a, b, c, d }` .
Long lists may be shortened with ellipses as long as the context is clear. For example, `E = { 2, 4, 6, 8, ... }` is clearly the set of all even numbers, containing an infinite number of objects, even though we've only explicitly written four of them.
### Set Builder
Set builder notation is a more descriptive way of constructing a set. It relies on a _subject_ and a _predicate_ such that `S = { subject : predicate }` . For example,
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```
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A = { x : x is a vowel } = { a, e, i, o, u, y}
B = { x : x ∈ N, x < 10 } = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 }
C = { x : x = 2k, k ∈ N } = { 0, 2, 4, 6, 8, ... }
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```
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Sometimes the predicate may "leak" into the subject, e.g.
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```
D = { 2x : x ∈ N } = { 0, 2, 4, 6, 8, ... }
```
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## Relations
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### Membership
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* If the value `a` is contained in the set `A` , then we say `a` belongs to `A` and represent this symbolically as `a ∈ A` .
* If the value `a` is not contained in the set `A` , then we say `a` does not belong to `A` and represent this symbolically as `a ∉ A` .
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### Equality
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* If two sets contain the same items then we say the sets are equal, e.g. `A = B` .
* Order does not matter when determining set equality, e.g. `{ 1, 2, 3, 4 } = { 2, 3, 1, 4 }` .
* Sets are disjoint, meaning elements cannot be repeated, e.g. `{ 1, 2, 2, 3, 4, 3, 4, 2 } = { 1, 2, 3, 4 }` .
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* Two sets `A` and `B` are equal if and only if `A ⊆ B` and `B ⊆ A` .
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## Special Sets
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### The Power Set
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* Let `A` be any set. The set that contains all possible subsets of `A` is called a "power set" and is written as `P(A)` . If the set `A` contains `n` elements, then `P(A)` contains `2^n` elements.
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```
P(A) = { x : x ⊆ A }
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```
## Set operations among two sets
### Union
Given two sets `A` and `B` , the union of the two sets are the items that appear in either `A` or `B` , written as `A ∪ B` .
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```
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A ∪ B = { x : x ∈ A ∪ x ∈ B }
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```
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### Intersection
Given two sets `A` and `B` , the intersection of the two sets are the items that appear in both `A` and `B` , written as `A ∩ B` .
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```
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A ∩ B = { x : x ∈ A, x ∈ B }
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```
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### Difference
Given two sets `A` and `B` , the set difference of `A` with `B` is every item in `A` that does not belong to `B` .
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```
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A \ B = { x : x ∈ A, x ∉ B }
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```
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### Symmetrical difference
Given two sets `A` and `B` , the symmetrical difference is all items among `A` and `B` that doesn't appear in their intersections.
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```
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A △ B = { x : ((x ∈ A) ∩ (x ∉ B)) ∪ ((x ∈ B) ∩ (x ∉ A)) }
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A △ B = (A \ B) ∪ (B \ A)
```
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### Cartesian product
Given two sets `A` and `B` , the cartesian product between `A` and `B` consists of a set containing all combinations of items of `A` and `B` .
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```
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A × B = { (x, y) | x ∈ A, y ∈ B }
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```