#### description

- Subsets of a Set of Real Numbers Especially Intervals - With Notation
- Singleton Set
- Super Set
- Subsets of set of real numbers
- Intervals as subsets of R

#### definition

A 'set A' is said to be a subset of a set B if every element of A is also an element of B.

#### notes

Sub means 'part of', thus subset means part of set.

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

A set is the part of set B

A is subset of B

Mathematically it is written as A ⊂ B

'⊂' means is subset of

B= {1, 2, 3, 4, 5}, A= {1, 2}, C= {3, 4}, D= {3, 5} and E= {1,5}

A, C, D and E are subsets of B

Mathematically, if A ⊂ B, a ∈ A ⇒ a ∈ B

'a' is an element of set A

'⇒' means implies

If A is not the subset of B, then mathematically we will write A ⊄ B

'⊄' means not the subset of

Case 1- If every elemento of A is also an element of B, i.e A ⊂ B

But if every element of B is also an element of A, i.e B ⊂ A

then A=B

A ⊂ B and B ⊂ A ⇔ A= B

'⇔' means two way implication

Note: 1) A = A

A⊂ A. Every set is subset of itself.

2) Ø ⊂ any set . Empty set is subset of any set.

**There are many important subsets of R (set of all real numbers). We give below the names of some of these subsets. **

The set of natural numbers N = {1, 2, 3, 4, 5, . . .}

The set of integers Z = {. . ., –3, –2, –1, 0, 1, 2, 3, . . .}

The set of rational numbers Q = { x : x = p/q, p, q ∈ Z and q ≠ 0}

The set of irrational numbers, denoted by T, is composed of all other real numbers. Thus T = {x : x ∈ R and x ∉ Q}, i.e., all real numbers that are not rational.

Members of T include 2 , 5 and π .

Some of the obvious relations among these subsets are:

N ⊂ Z ⊂ Q, Q ⊂ R, T ⊂ R, N ⊄ T.

Let A & B be two sets. A ⊂ B and A ≠ B.

Then A is called proper subset. and B is called **superset**.

A set containing only one element is known as **Singleton set**.

A= {1}, B= {Ø}, C= {2}

**Intervals as subset of R**

If A= {x: 5 ≤ x ≤ 9 ∀ x ∈ R}

Real Interval [5,9] is subset of R, here 5 and 9 are included therefore we wrote them in these '[ ]' brackets and such intervals are known as closed intervals.

If B= {x:5 < x < 9, ∀ x ∈ R}

Real Interal (5,9) is subset of R, here 5 and 9 are excluded therefore we wrote them in these '( )' brakets and such intervals are known as open intervals.

If C= {x: 5 < x ≤ 9 ∀ x ∈ R}

Real interval (5,9] is subset of R, here 5 is excluded and 9 is included therefore we wrote them in '(' and ']' brackets such intervals are known as open closed intervals.

If D= {x: 5 ≤ x < 9, ∀ x ∈ R}

Real interval [5,9) is subset of R, here 5 is included and 9 is excluded therefore we wrote them in '[' and ')' brackets such intervals are known as closed open intervals.

**Length of Interval:** if [a,b], (a,b), [a,b), (a,b] are the subsets, then b-a is the length of interval.