**Basics of Set Theory**

Set Theory is considered as a

part of modern mathematics that helps in developing a fruitful research area of

its own. Set theory had its origin in work done by Georg Cantor in the late 19^{th}

century mainly on certain kinds of infinite series called Fourier series. From

this point of view, most mathematicians accept set theory as a foundation for

mathematics. The set notions and the membership in a set can be used as the

most primitive ideas in terms of which all-mathematical objects and ideas can

be defined.

**Basic Concepts (Signs/
Notation/ Explanation)**

Set Theory begins with a

fundamental binary relation between an object and set. For instance if A is a

set and x is an element of A, we write x â‚¬ A. A binary relation between two set

can also be a subset relation also called Set inclusion. This happens when all

the members of set A are also members of set B, then A is the subset of B, well

denoted by A Â B. For instance, {1,2} is

a subset of {1,2,3} but {1,4} is not. Hence, right from this definition it is

clear that a set is the subset of itself.

Two sets are said to be equal if

and only if they have the same elements. Thus for example {1, 2, 3} = {3, 2, 1}

that means the order of elements does not matter.

Similarly, in understanding subsets,

set A is a subset of a set B when everything** **in A is also in B. More

properly, for sets A and B, A is a proper subset of B and denoted by A Â B, when

Â A Â B

In cardinality, if a set S has n

distinct element for some natural number n, then n is the cardinality of S and

S is a finite set. For example, the cardinality of the set {*3, 1, 2*} is 3.

Empty or null set needs no

mentioning. More formally, an empty set is denoted by Ã˜.Â Note that Ã˜ and {Ã˜}

are different sets. {Ã˜} has one element namely Ã˜ in it. So, {Ã˜} is not empty.

But, Ã˜ has nothing in it.

Universal set is a set, which has

all the elements in the universe of discourse. More properly, a universal set

denoted by, denoted by U.

**Rules**

Letâ€™s say that our universe contains

the numbers 1, 2, 3, and 4. Let A be the set containing the numbers 1 and 2,

that is A= {1, 2} and let B be the set containing the numbers 2 and 3; that is B

= {2, 3}. Here are the following relationships with blue shade, marking the

solution region in the Venn diagrams.

If the set notation is A U B then

it would be everything that falls in either of the sets. Here the answer will

be {1, 2, 3}

If the set notation is A

intersection B or Aâˆ©B, it means the notation is referring to the elements

that are common in both of the sets. Here the answer will be {2}

Â

If the set notation is ~ A. This

means all elements in the universe outside of A. Here the answer will be {3,

4}.

Â

If the set notation is A- B, (i.e

A minus B or A compliment B), it means everything in A except for anything in

its overlap with B. Here the answer will be {1}.

If the set notation is ~ (A U B),

i.e not (A union B). This denotes everything that falls outside A and B. Here

the answer will be {4}.

If the set notation is ~ (A ^ B)

it means everything outside the overlapping region of A and B giving answer

like {1, 3, 4}.

Â

As you can see above a subset is

a set, which is totally contained within another set. For instance, every set

in a Venn Diagram is a subset of the diagramâ€™s universe.

Venn diagram can also demonstrate

disjoint sets. In the above graphic representation, you will find A and B are

disjoint. As disjoint sets have no overlapping, so this means their

intersection is empty.

However, we can follow some other

ways to describe sets. When the set is a small finite set or an infinite set

whose elements can be referred to using “â€¦” For example

A= {1, 2}

N= {1, 2, 3â€¦} These are all set

of natural numbers. Again, there are some people who include “0” in what they

call the set of natural numbers. W= {0, 1, 2â€¦} that are known as the set of

whole numbers while Z = {0, Â±1, Â± 2…}. These are known as set of integers.

**Concepts like ****Union****,
Intersections and compliments **

Just as arithmetic features

binary operations on numbers, set theory features binary operations on sets.

– Union of set A and B denoted AU

B is the set whose members are members of at least one of A or B. The union of

{1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}

– Intersection of sets A and B, denoted

by A âˆ© B is the set whose members are both A and B. The intersection of

{1, 2, 3} and {2, 3, 4} is the set {2, 3}

– Compliment of set A relative to Set U denoted by A^{ c}

is the set of all members of U that are not members of A. This term is commonly

employed when U is a universal set. This operation is also called the set

difference of U and A, denoted by U \ A. For example the compliment of {1,2,3}

relative to {2,3,4}is {4} while the compliment of {2,3,4} relative to {1,2,3}

is {1}.