sets & set theory | relations & functions

Описание: sets & set theory | relations & functions #mathematics #maths #algebra

Definition of Sets
Sets are represented as a collection of well-defined objects or elements and it does not change from person to person. A set is represented by a capital letter. The number of elements in the finite set is known as the cardinal number of a set.

N: Set of all natural numbers
Z: Set of all integers
Q: Set of all rational numbers
R: Set of all real numbers
Z+: Set of all positive integers

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Order of Seta
The order of a set defines the number of elements a set is having. It describes the size of a set. The order of set is also known as the cardinality.

Representation of Sets
The sets are represented in curly braces, {}. For example, {2,3,4} or {a,b,c} or {Bat, Ball, Wickets}. The elements in the sets are depicted in either the Statement form, Roster Form or Set Builder Form.

Statement Form
In statement form, the well-defined descriptions of a member of a set are written and enclosed in the curly brackets.

Roster Form
In Roster form, all the elements of a set are listed
Set Builder Form
The general form is, A = { x : property }

Empty Set
A set which does not contain any element is called an empty set or void set or null set. It is denoted by { } or Ø.

A set of apples in the basket of grapes is an example of an empty set because in a grapes basket there are no apples present.

Singleton Set
A set which contains a single element is called a singleton set.

Finite set
A set which consists of a definite number of elements is called a finite set.

Infinite set
A set which is not finite is called an infinite set.

Equivalent set
If the number of elements is the same for two different sets, then they are called equivalent sets. The order of sets does not matter here. It is represented as : n(A) = n(B)

Equal sets
The two sets A and B are said to be equal if they have exactly the same elements, the order of elements do not matter.

Disjoint Sets
The two sets A and B are said to be disjoint if the set does not contain any common element.

Subsets
A set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted as A ⊆ B. Even the null set is considered to be the subset of another set. In general, a subset is a part of another set.
Note: The set is also a subset of itself.
If A is not a subset of B, then it is denoted as A⊄B.

Proper Subset
If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B.

Superset
Set A is said to be the superset of B if all the elements of set B are the elements of set A. It is represented as A ⊃ B

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