 # Set Theory in Maths

Set theory was developed to explain about collections of objects, in Maths. Basically, the definition states it is a collection of elements. These elements could be numbers, alphabets, variables, etc. The notation and symbols for sets are based on the operations performed on them. You must have also heard of subset and superset, which are the counterpart of each other.

Sets have turned out to be an invaluable tool for defining some of the most complicated mathematical structures. They are mostly used to define many real-life applications. Apart from this, there are also many types of sets, such as empty set, finite and infinite set, etc. These are explained widely with the help of Venn diagrams.

## Sets in Maths

As we have already discussed, sets is a collection for different types of objects and collectively itself is called an object. For example, number 8, 10, 15, 24 are 4 distinct numbers, but when we put them together, they form a set of 4 elements, such that, {8, 10, 15, 24}.

In the same way, sets are defined in the Maths for a different pattern of numbers or elements. Such as, sets could be a collection of odd numbers, even numbers, natural numbers, whole numbers, real or complex numbers and all the set of numbers which comes in the number line.

## Symbols Used For Representation of Sets

Let us see the different types of symbols we used while we learn about sets. Consider a Universal set (U) = {1, 2, 7, 9, 13, 15, 21, 23, 28, 30}

##### Example
{ } set a collection of elements A = {1, 7, 9, 13, 15, 23},

B = {7, 13, 15, 21}

A ∪ B union objects that belong to set A or set B A ∪ B = {1, 7, 9, 13, 15, 21, 23}
A ∩ B intersection objects that belong to both the sets, A and B A ∩ B = {7, 13, 15 }
A ⊆ B subset subset has few or all elements equal to the set {7, 15} ⊆ {7, 13, 15, 21}
A ⊄ B not subset left set not a subset of right set {1, 23} ⊄ B
A ⊂ B proper subset / strict subset subset has fewer elements than the set {7, 13, 15} ⊂ {1, 7, 9, 13, 15, 23}
A ⊃ B proper superset / strict superset set A has more elements than set B {1, 7, 9, 13, 15, 23} ⊃ {7. 13. 15. }
A ⊇ B superset set A has more elements or equal to the set B {1, 7, 9, 13, 15, 23} ⊃ {7. 13. 15. 21}
Ø empty set Ø = { } C = {Ø}
P (C) power set all subsets of C C = {4,7},

P(C) = {{}, {4}, {7}, {4,7}}

Given by 2s, s is number of elements in set C

A ⊅ B not superset set A is not a superset of set B {1, 7, 9, 13, 15, 23} ⊅{7, 13, 15, 21}
A = B equality both sets have the same members {7, 13,15} = {7, 13, 15}
A \ B or A-B relative complement objects that belong to A and not to B {1, 9, 23}
Ac complement all the objects that do not belong to set A We know, U = {1, 2, 7, 9, 13, 15, 21, 23, 28, 30}

Ac = {2, 21, 28, 30}

A ∆ B symmetric difference objects that belong to A or B but not to their intersection A ∆ B = {1, 9, 21, 23}
a∈B element of set membership B = {7, 13, 15, 21},

13 ∈ B

(a,b) ordered pair collection of 2 elements
x∉A not element of no set membership A = {1, 7, 9, 13, 15, 23, 5 ∉ A
|B|, #B cardinality the number of elements of set B B = {7, 13, 15, 21}, |B|=4
A×B cartesian product set of all ordered pairs from A and B {3,5} × {7,8} = {(3,7), (3,8), (5,7), (5, 8) }
N1 natural numbers / whole numbers  set (without zero) N1 = {1,2,3,4,5,…} 6 ∈ N1
N0 natural numbers / whole numbers  set (with zero) N0 = {0,1,2,3,4,…} 0 ∈ N0
Q rational numbers set Q= {x | x=a/b, a,b∈Z} 2/6 ∈ Q
Z integer numbers set Z= {…-3,-2,-1,0,1,2,3,…} -6 ∈ Z
C complex numbers set C= {z | z=a+bi, -∞<a<∞,                         -∞<b<∞} 6+2i ∈ C
R real numbers set R= {x | -∞ < x <∞} 6.343434 ∈ R

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