 # Set Theory in Maths

In Maths, the Set theory was developed to explain about collections of objects. Basically, the definition states that “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. The different types of sets in Mathematics set theory are explained widely with the help of Venn diagrams.

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.

## What is Set Theory in Maths?

As we have already discussed, in mathematics set theory, a set is a collection for different types of objects, and collectively itself is called an object. For example, number 8, 10, 15, 24 are the 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.

### History

A Greek mathematician Georg Cantor generated a theory of abstract sets of entities and formed it into a mathematical discipline between the years 1874 and 1897. This theory in maths built out of his research of some definite problems about specific types of infinite sets of numbers which are real. According to Cantor, the set is a collection of definite, distinct objects or items of observation as a whole. These items are called elements or members of the set. However, he founded it by a single paper based on the property of the combination of all real numbers (or real algebraic numbers).

## Mathematics Set Theory Symbols

Let us see the different types of symbols used in Mathematics set theory with its meaning and examples.  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 Elements that belong to set A or set B A ∪ B = {1, 7, 9, 13, 15, 21, 23}
A ∩ B intersection Elements 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 is 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 X is not a superset of set Y {1, 2, 5} ⊅{1, 6}
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 (1, 2)
x∉A not element of no set membership A = {1, 7, 8, 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

### Basic Concepts of Set Theory

In set theory, various concepts are discussed at various levels of education. The basic concepts out of which include representation of a set, types of sets, operations on sets (such as union, intersection), cardinality of a set and relations, etc.

### Applications

Set theory has many applications in mathematics and other fields. They are used in graphs, vector spaces, ring theory, and so on. All these concepts can be defined as sets satisfying specific properties (or axioms) of sets. Also, the set theory is considered as the foundation for many topics such as topology, mathematical analysis, discrete mathematics, abstract algebra, etc.

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