Arithmetic Sets

Set is collection of distinct objects

∅ - is null set

Membership

A = {1,2,3,4}
4 ∈ A 
9 ∉ A 

Equality

A is {1, 2, 3}
B is {3, 1, 2}

A = B

Unions

Two sets can be “added” together. The union of A and B, denoted by A ∪ B, is the set of all things that are members of either A or B.

{1, 2} ∪ {1, 2} = {1, 2}
{1, 2} ∪ {2, 3} = {1, 2, 3}
{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}

Intersections

{1, 2} ∩ {1, 2} = {1, 2}
{1, 2} ∩ {2, 3} = {2}

Complements

Two sets can also be “subtracted”. The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B)

A′ = U \ A

{1, 2} \ {1, 2} = ∅.
{1, 2, 3, 4} \ {1, 3} = {2, 4}.

A \ B ≠ B \ A  for A ≠ B.

Universal set and absolute complement

U  - Universal set
A′ - All things in the universe that are not in A

A′ = U \ A = U - A

C = {0,5,15,3}

0 ∈ C 
12 ∉ C

C′ = U\C = U-C

0 ∉ C′ 
12 ∈ C′

Subsets

If every member of set A is also a member of set B, then A is said to be a subset of B, written A ⊆ B (also pronounced A is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment.

subset of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc.

{1, 3} ⊆ {1, 2, 3, 4}.
{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The empty set is a subset of every set and every set is a subset of itself:

∅ ⊆ A.
A ⊆ A.

Every set is a subset of the universal set:

A ⊆ U.

number of subset of set is 2^n
n is number of elements

Proper Subsets (strict subsets)

If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ⊊ B (A is a proper subset of B) or B ⊋ A (B is a proper superset of A).

{1, 2, 3} is a subset of {1, 2, 3}, 
but is not a proper subset of {1, 2, 3}.

{1, 2, 3} is a proper subset of {1, 2, 3, 4}
because the element 4 is not in the first set.

{1, 2, 3, 4}  is a proper superset of {1, 2, 3}

{1, 2, 3} ⊆ {1, 2, 3}
{1, 2, 3} ⊊ {1, 2, 3, 4} we can say {1, 2, 3} ⊆ {1, 2, 3, 4}
{1, 2, 3, 4} ⊋ {1, 2, 3} 

Posibile Subset

{1, 2, 3}
subsets
{} or ∅,{1},{2},{3},{1,2},{1,3},{2,3},{1, 2, 3}

References

• https://en.wikipedia.org/wiki/Set_(mathematics)/