`star` Intervals as subsets of R

`star` Power Set

`star` Universal Set

`star` Power Set

`star` Universal Set

`\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].`

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

(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

(i) `(-4,6]`

(ii) `(-12,-10 )`

(iii) `[0 , 7)`

(iv) `[3,4]`

`"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`

• 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 }.`

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 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.

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

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

(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`.