Set Operators／Operations

{ }

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: 𝐴 has some (or all) elements of 𝐵

{3,4,5} ⊆ 𝐷

𝐴 ⊂ 𝐵

Proper Subset: 𝐴 has some elements of 𝐵

{3,5} ⊂ 𝐷

𝐴 ⊄ 𝐵

Not a Subset: 𝐴 is not 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 is not a superset of 𝐵

{1,2,6} ⊅ {1,9}

𝐴^𝑐

Complement: elements not in 𝐴

𝐷^𝑐 = {1,2,6,7}
When 𝕌 = {1,2,3,4,5,6,7}

𝐴 − 𝐵 or 𝐴\𝐵

Set Difference (Relative Complement): in 𝐴 but not in 𝐵

{1,2,3,4} − {3,4} = {1,2}

Symmetric Difference

a ∈ 𝐴

Element of: 𝑎 is in 𝐴

3 ∈ {1,2,3,4}

b ∉ 𝐴

Not element of: 𝑏 is not in 𝐴

6 ∉ {1,2,3,4}

∅

Empty set = {}

{1,2} ∩ {3,4} = Ø

𝕌

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

∃

There Exists

∃𝑥 | 𝑥²>𝑥

∴

Therefore

𝑎=𝑏 ∴ 𝑏=𝑎