Sets introduction Definations | Maths EM

Object : Everything in this universe whether living or non living is called an object.

Set : – A set is a collection of well-defined objects.
A set is a collection of things that have something in common or following a rule.

Well-defined means, it must be absolutely clear that which object belongs to the set and which does not.
Elements of Set:
The things or objects in the set are called “ elements “

Notation of a Set: A set is usually denoted by capital letters and elements are denoted by small letters

The elements of set are written inside a pair of curly braces separated by commas.
The elements of a set may be written in any order.
The elements of a set must not be repeated.
The Greek letter Epsilon ‘∈’ is used for the words ‘belongs to’, ‘is an element of’, etc.
Therefore, x ∈ A will be read as ‘x belongs to set A’ or ‘x is an element of the set A’.
The symbol ‘∉’ stands for ‘does not belongs to’ also for ‘is not an element of’.

Ex: – The collection of positive numbers less than 10’ is a set.
The collection of good students in your class’ is not a set.
The collection of first five months of a year is a set.
The collection of rich man in your town is not a set.
The collection of vowels in English alphabets. This set contains five elements,namely a, e, i, o, u.

In representation of a set the following two methods are commonly used:
(i) Roster or tabular form method
(ii) Rule or set builder form method

Roster form or tabular form:
In this, elements of the set are listed within the pair of brackets { } and are separated by commas.

Singleton Set: A set which contains only one element is called a singleton set.
Ex A = {x : x is neither prime nor composite}

B = {x : x is a whole number, x < 1} is a singleton set containing one element, i.e., 1.
This set contains only one element 0 and is a singleton set.
• Let A = {x : x ∈ N and x² = 4}
Here A is a singleton set because there is only one element 2 whose square is 4.
• Let B = {x : x is a even prime number}
Here B is a singleton set because there is only one prime number which is even, i.e., 2.

UNIVERSAL SET:
A universal set is a set which contains all objects including itself.
All sets under consideration will be subsets of a fixed set. We call this set the Universal Set.
A set which contains all the elements of other given sets is called a universal set.
The symbol for denoting a universal set is or ξ.

Subset:- If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B, and we write it as A ⊂   B or B ⊂ A
If all elements of set A are present in B , then A is said to be subset of B denoted by A ⊂ B
The symbol ⊂  stands for ‘is a subset of’ or ‘is contained in’
• Every set is a subset of itself,
i.e., A ⊂ A, B ⊂ B.
• Empty set is a subset of every set.
• Symbol ‘⊆’ is used to denote ‘is a subset of’ or ‘is contained in’.
• A ⊆ B  means A is a subset of B or A is contained in B.
• B ⊆ A means B contains A.

For example;
1. Let A = {2, 4, 6} B = {6, 4, 8, 2} Here A is a subset
of B
2. Notes: If A⊂ B and B⊂ A, then A = B, i.e., they are equal sets.

Super Set: Let A and B are two sets. If A⊂ B , then B is called the super set of A.
Whenever a set A is a subset of set B, we say that B is a superset of A and we write, B ⊇ A.
Symbol ⊇ is used to denote ‘ is a super set of ’
For example;

A = {a, e, i, o, u} B = {a, b, c, …………., z}
Here A ⊆ B i.e., A is a subset of B but B ⊇ A i.e., B is a super set of A

In representation of a set the following two methods are commonly used: