Set Theory Symbols

Posted in engineering by Christopher R. Wirz on Wed Feb 08 2017

Set notation is an important convention in computer science. This cheat sheet is extremely useful.

Symbol Symbol Name Meaning /
definition
Example
{ } set a collection of elements A = {3,7,9,14},
B = {9,14,28}
| such that so that A = {x | x\mathbb{R}, x<0}
A∩B intersection objects that belong to set A and set B A ∩ B = {9,14}
A∪B union objects that belong to set A or set B A ∪ B = {3,7,9,14,28}
A⊆B subset subset has fewer elements or equal to the set {9,14,28} ⊆ {9,14,28}
A⊂B proper subset / strict subset subset has fewer elements than the set {9,14} ⊂ {9,14,28}
A⊄B not subset left set not a subset of right set {9,66} ⊄ {9,14,28}
A⊇B superset set A has more elements or equal to the set B {9,14,28} ⊇ {9,14,28}
A⊃B proper superset / strict superset set A has more elements than set B {9,14,28} ⊃ {9,14}
A⊅B not superset set A is not a superset of set B {9,14,28} ⊅ {9,66}
2A power set all subsets of A  
\mathcal{P}(A) power set all subsets of A  
A=B equality both sets have the same members A={3,9,14},
B={3,9,14},
A=B
Ac complement all the objects that do not belong to set A  
A\B relative complement objects that belong to A and not to B A = {3,9,14},
B = {1,2,3},
A \ B = {9,14}
A-B relative complement objects that belong to A and not to B A = {3,9,14},
B = {1,2,3},
A - B = {9,14}
A∆B symmetric difference objects that belong to A or B but not to their intersection A = {3,9,14},
B = {1,2,3},
A ∆ B = {1,2,9,14}
A⊖B symmetric difference objects that belong to A or B but not to their intersection A = {3,9,14},
B = {1,2,3},
A ⊖ B = {1,2,9,14}
a∈A element of set membership A={3,9,14}, 3 ∈ A
x∉A not element of no set membership A={3,9,14}, 1 ∉ A
(a,b) ordered pair collection of 2 elements  
A×B cartesian product set of all ordered pairs from A and B  
|A| cardinality the number of elements of set A A={3,9,14}, |A|=3
#A cardinality the number of elements of set A A={3,9,14}, #A=3
symbol aleph-null infinite cardinality of natural numbers set  
symbol aleph-one cardinality of countable ordinal numbers set  
Ø empty set Ø = {} A = Ø
\mathbb{U} universal set set of all possible values  
\mathbb{N}0 natural numbers / whole numbers  set (with zero) \mathbb{N}0 = {0,1,2,3,4,...} 0 ∈ \mathbb{N}0
\mathbb{N}1 natural numbers / whole numbers  set (without zero) \mathbb{N}1 = {1,2,3,4,5,...} 6 ∈ \mathbb{N}1
\mathbb{Z} integer numbers set \mathbb{Z} = {...-3,-2,-1,0,1,2,3,...} -6 ∈ \mathbb{Z}
\mathbb{Q} rational numbers set \mathbb{Q} = {x | x=a/b, a,b\mathbb{Z}} 2/6 ∈ \mathbb{Q}
\mathbb{R} real numbers set \mathbb{R} = {x | -∞ < x <∞} 6.343434 ∈ \mathbb{R}
\mathbb{C}  complex numbers set \mathbb{C} = {z | z=a+bi, -∞<a<∞,      -∞<b<∞} 6+2i\mathbb{C}
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