Introduction To Sets

Introduction To Sets

A group or collection of unique, well-defined objects is referred to as a set. 

For instance, a box of pencils is a set.

Consider a "collection of interesting stories" at this point. 

Different people have different ideas of what "interesting stories" are. 

This phrase is not specifically defined. 

Therefore, it doesn't make up a set. 

A set of items is written with two curly brackets around them.
Members or elements of the set are what we refer to as these items. It
is said that every component belongs to the set. The word "E" stands for "belongs to," is an element of," or "is a member of . 
"A set of ice cream flavours offered in an ice cream shop is F, for  example. .

How is a set notation?

Three ways can be used to represent a set:

• Descriptive form:

A set described in descriptive terms is in the descriptive form. 

For instance:

A  first five natural integers

• Set builder form:

A set is written as a rule

Example:

 A = {x: x is a natural number and x<10}

● Tabular form:

A set with the set's components written in plain language.

Example: 

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

                                                                Types of sets

Empty set:

A set holds no elements. It is indicated by { } or 0.

Example: 

B = {integers between 0 and 1} or B = {}

Finite set:

A set with a limited number of elements

Example: 

C= {x|x is a square number , x < 20} or C = {1, 4, 9, 16}

Infinite set:

A set that is not finite.

Example: 

A = {x: x is an integer, x>10} or A = {11, 12, 13, 14, 15, ......}

Disjoint sets:

Sets without any shared elements.

Example:

 A = {a, i, o} and B = {c, d, e}

Overlapping sets:

Sets that share at least one element.

Example: 

C = {1, 2, 3) and D = {2, 3, 4}

Equivalent sets:

Sets with the same amount of elements.
The elements of equivalent sets don’t have to be exactly the same.
Example: E = {1, 4, 6) and F = {a, b, c) Consider, A = {planets in the solar
system) and B = {natural numbers less than 9}
Both sets A and B have 8 members each, even though the members are
completely different. These sets are equivalent sets.

Equal sets:

Sets containing the same elements.

Example: 

L = {2, 4, 6} and M = {x: x is a positive multiple of 2, x<7).

Consider, A = {even natural numbers) and B = {multiples of 2}. Both sets

A and B contain the elements 2, 4, 6, and 8, .... Hence, they are equal sets.

Universal set:

A set that contains each set being considered.

Example:

 If J = {January, July, June) and A = {April, August} are the sets

under consideration,

then U = {months of the year} is the universal set.

Subset:

A  bigger set enclosing a smaller one from all sides. The larger set is called the superset.

Example:

 A = {a, e, i} is a subset of B = {a, e, i, o, u) and B is the 

superset of A. 

It is depicted as A ⊂B (A is a subset of B) and B ⊃A (B is the 

superset of A)

Example:

 A = {2, 4, 6}

Then, subsets of A are: { }, {2}, {4}, {6}, {2, 4}, {2, 6}, {4,6}, {2, 4, 6} Null

A Set is a subset of every set and each set is a subset of itself. All other

sets are called proper subsets.

There is 1 - 1 correspondence between the two sets when each

member of one set matches one and only one member of the other set.

Equivalent sets are those that may be paired in a one-to-one

relationship.

For instance, every element in set B has a corresponding element in the set

A.

A = {1, 2, 3}, B = {7, 14, 21}

Non-equivalent sets are those that cannot be paired in a one-to-one

relationship.

Cardinal number of a set:

The number of elements of a set is called its cardinal number.