Mathematics Subsets , Intervals as subsets of R , Power Set and Universal Set
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### Topics Covered

star Intervals as subsets of R
star Power Set
star Universal Set

### Intervals as subsets of R

\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)
Let a, b ∈ R and a < b.

• Then the set of real numbers { y : a < y < b} is called an "open interval" and is denoted by (a, b).

• All the points between a and b belong to the open interval (a, b) but a, b themselves do not belong to this interval.

• The interval which contains the end points also is called "closed interval" and is denoted by [ a, b ]. Thus
 \ \ \ \ \ \ \ \ \ [ a, b ] = {x : a ≤ x ≤ b}

• We can also have intervals closed at one end and open at the other, i.e.,
[ a, b ) = {x : a ≤ x < b} is an open interval from a to b, including a but excluding b.
( a, b ] = { x : a < x ≤ b } is an open interval from a to b including b but excluding a.

These intervals are also known as semi open or semi closed or half open or half closed.
[a,b) is also written as  [a,b[

\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)
(a,b] is also written as  ]a,b]

• These notations provide an alternative way of designating the subsets of set of real numbers.
For example , if A = (–3, 5) and B = [–7, 9], then A ⊂ B.

On real number line, various types of intervals described above as subsets of R, are shown in the Fig.

• Here, we note that an interval contains infinitely many points.
The number (b – a)  is called the length of any of the intervals (a, b), [a, b], [a, b) or (a, b].
Q 1904145958

Write the following as intervals :

(i) {x : x in R , - 4 < x le 6}

(ii) {x : x in R, - 12 < x < -10}

(iii) {x : x in R, 0 le x < 7}

(iv) {x : x in R, 3 le x le 4}
Class 11 Exercise 1.3 Q.No. 6
Solution:

(i) (-4,6]

(ii) (-12,-10 )

(iii) [0 , 7)

(iv) [3,4]

### Power Set :

"Definition :" The collection of all subsets of a set A is called the power set of A.
• It is denoted by P(A). In P(A), every element is a set.

E.g. If A = { 1, 2 }, then P( A ) = { Phi ,{ 1 }, { 2 }, { 1,2 }} Also, note that n [ P (A) ] = 4 = 2^2

In general, if A is a set with n(A) = m, then it can be shown that n [ P(A)] = 2^m
Q 3087578487

List all the subsets of the set { –1, 0, 1 }.

Solution:

Let A = { –1, 0, 1 }. The subset of A having no element is the empty
set phi. The subsets of A having one element are { –1 }, { 0 }, { 1 }. The subsets of
A having two elements are {–1, 0}, {–1, 1} ,{0, 1}. The subset of A having three
elements of A is A itself. So, all the subsets of A are phi, {–1}, {0}, {1}, {–1, 0}, {–1, 1},
{0, 1} and {–1, 0, 1}.

### Universal Set :

• Universal Set is the set defines as the set containing all objects or elements and of which all other sets are subsets.

or

• A universal set is the collection of all objects in a particular context or theory. All other sets in that framework constitute subsets of the universal set.

• Universal Sets are usually named with a capital letter. Therefore, the universal set is usually named with the capital letter U.

For example, R can be set of real numbers.
Q 1964256155

Given the sets A = { 1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be

considered as universal set (s) for all the three sets A, B and C.

(i) {0, 1, 2, 3, 4, 5, 6}

(ii) {phi}

(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(iv) {1,2,3,4,5,6,7,8}
Class 11 Exercise 1.3 Q.No. 9
Solution:

(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} may be

considered-as universal set for all the three sets

A, B and C.