Set Symbols
A set is a collection of things, usually numbers. We can list each element (or "member") of a set inside curly brackets like this:
Common Symbols Used in Set Theory
Symbols save time and space when writing. Here are the most common set symbols:
In the examples C = {1, 2, 3, 4} and D = {3, 4, 5}
| Symbol | Meaning | Example |
|---|---|---|
| { } | Set: a collection of elements | {1, 2, 3, 4} |
| A ∪ B | Union: in A or B (or both) | C ∪ D = {1, 2, 3, 4, 5} |
| A ∩ B | Intersection: in both A and B | C ∩ D = {3, 4} |
| A ⊆ B | Subset: every element of A is in B. | {3, 4, 5} ⊆ D |
| A ⊂ B | Proper Subset: every element of A is in B, but B has more elements. |
{3, 5} ⊂ D |
| A ⊄ B | Not a Subset: A isn't a subset of B | {1, 6} ⊄ C |
| A ⊇ B | Superset: A has same elements as B, or more | {1, 2, 3} ⊇ {1, 2, 3} |
| A ⊃ B | Proper Superset: A has B's elements and more | {1, 2, 3, 4} ⊃ {1, 2, 3} |
| A ⊅ B | Not a Superset: A isn't a superset of B | {1, 2, 6} ⊅ {1, 9} |
| Ac | Complement: elements not in A | Dc = {1, 2, 6, 7} When = {1, 2, 3, 4, 5, 6, 7} |
| A − B | Difference: in A but not in B | {1, 2, 3, 4} − {3, 4} = {1, 2} |
| a ∈ A | Element of: a is in A | 3 ∈ {1, 2, 3, 4} |
| b ∉ A | Not element of: b isn't in A | 6 ∉ {1, 2, 3, 4} |
| ∅ | Empty set = {} | {1, 2} ∩ {3, 4} = ∅ |
| Universal Set: set of all possible values (in the area of interest) |
||
| P(A) | Power Set: all subsets of A | P({1, 2}) = { {}, {1}, {2}, {1, 2} } |
| A = B | Equality: both sets have the same members | {3, 4, 5} = {5, 3, 4} |
| A×B | Cartesian Product (set of ordered pairs from A and B) |
{1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)} |
| |A| | Cardinality: the number of elements of set A | |{3, 4}| = 2 |
| | | Such that | { n | n > 0 } = {1, 2, 3,...} |
| : | Such that | { n : n > 0 } = {1, 2, 3,...} |
| ∀ | For All | ∀x>1, x2>x For all x greater than 1 x-squared is greater than x |
| ∃ | There Exists | ∃ x | x2>x There exists x such that x-squared is greater than x |
| ∴ | Therefore | a=b ∴ b=a |
| Natural Numbers | {1, 2, 3,...} or {0, 1, 2, 3,...} | |
| Integers | {..., −3, −2, −1, 0, 1, 2, 3, ...} | |
| Rational Numbers | 34, 0.5, −2 | |
| Algebraic Numbers | ||
| Real Numbers | π, √2, 1.5, 7, −8 | |
| Imaginary Numbers | 3i | |
| Complex Numbers | 2 + 5i |