`star` The Empty Set

`star` Finite and Infinite Sets & Cardinality of a Finite Set

`star` Equal Sets

`star` Finite and Infinite Sets & Cardinality of a Finite Set

`star` Equal Sets

`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"`

A set which does not contain any element is called the `"empty set"` or the `"null set"` or the `"void set"`.

`\color{green} ✍️ ` The empty set is denoted by the symbol `\color{blue}(Phi)` or `\color{blue}({ }).`

`color{green}(E.g.)`

(i) Let `A =` {`x : 1 < x < 2, x` is a natural number}. Here is no natural number between 1 and 2. So `A` is the empty set.

(ii) `B =` {`x : x^2 – 2 = 0` and `x` is rational number}. Then `B` is the empty set because the equation `x^2 – 2 = 0` is not satisfied by any rational value of `x`.

A set which does not contain any element is called the `"empty set"` or the `"null set"` or the `"void set"`.

`\color{green} ✍️ ` The empty set is denoted by the symbol `\color{blue}(Phi)` or `\color{blue}({ }).`

`color{green}(E.g.)`

(i) Let `A =` {`x : 1 < x < 2, x` is a natural number}. Here is no natural number between 1 and 2. So `A` is the empty set.

(ii) `B =` {`x : x^2 – 2 = 0` and `x` is rational number}. Then `B` is the empty set because the equation `x^2 – 2 = 0` is not satisfied by any rational value of `x`.

Q 1904734658

Which of the following are examples of the null set:

(i) Set of odd natural numbers divisible by `2`.

(ii) Set of even prime numbers.

(iii) `{ x : x` is a natural numbers , `x < 5` and `x > 7} `

(iv) `{ y : y` is a point common to any two parallel lines}

Class 11 Exercise 1.2 Q.No. 1

(i) Set of odd natural numbers divisible by `2`.

(ii) Set of even prime numbers.

(iii) `{ x : x` is a natural numbers , `x < 5` and `x > 7} `

(iv) `{ y : y` is a point common to any two parallel lines}

Class 11 Exercise 1.2 Q.No. 1

(i) Set of odd natural numbers divisible by `2` is a null set, because no odd natural number is divisible by `2`.

(ii) We know that `2` is the only even prime number. Therefore, set of even prime numbers is not a null set .

(iii) `{ x : x in N, x < 5` and `x > 7 }` is a null set, because there is no natural number which is less than `5` and greater than `7` simultaneously.

(iv) `{ v : y` is a point common to any two parallel lines} is a nuII set, because there is no point common to any two parallel lines.

`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"`

A set which is empty or consists of a definite number of elements is called `"finite"` otherwise, the set is called `"infinite."`

`\color{green} "Consider some examples :"`

`(i)` Let `W` be the set of the days of the week. Then `W` is finite.

`(ii)` Let `S` be the set of solutions of the equation `x^2 –16 = 0`. Then `S` is finite.

`(iii)` Let `G` be the set of points on a line. Then `G` is infinite.

`\color{green} ✍️ \color{green} \mathbf(KEY \ POINTS)`

• When we represent a set in the roster form, we write all the elements of the set within braces `{ }.`

• It is not possible to write all the elements of an infinite set within braces `{ }` because the numbers of elements of such a set is not finite.

• So, we represent some infinite set in the roster form by writing a few elements which clearly indicate the structure of the set followed ( or preceded ) by three dots.

`"For example,"` `{1, 2, 3 . . .}` is the set of natural numbers,

` \ \ \ \ \ \ \ \ \ \ \ {1, 3, 5, 7, . . .} ` is the set of odd natural numbers,

` \ \ \ \ \ \ \ \ \ \ \ {. . .,–3, –2, –1, 0,1, 2 ,3, . . .}` is the set of integers. All these sets are infinite.

`\color{green} ✍️ \color{green} \mathbf(KEY \ POINTS)`

• All infinite sets cannot be described in the roster form.

• For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

`\color{green} ✍️ \color{green} \mathbf( "Cardinality of a finite set" )`

`"Cardinality"` of a finite Set is defined as total number of elements in a set.

A set which is empty or consists of a definite number of elements is called `"finite"` otherwise, the set is called `"infinite."`

`\color{green} "Consider some examples :"`

`(i)` Let `W` be the set of the days of the week. Then `W` is finite.

`(ii)` Let `S` be the set of solutions of the equation `x^2 –16 = 0`. Then `S` is finite.

`(iii)` Let `G` be the set of points on a line. Then `G` is infinite.

`\color{green} ✍️ \color{green} \mathbf(KEY \ POINTS)`

• When we represent a set in the roster form, we write all the elements of the set within braces `{ }.`

• It is not possible to write all the elements of an infinite set within braces `{ }` because the numbers of elements of such a set is not finite.

• So, we represent some infinite set in the roster form by writing a few elements which clearly indicate the structure of the set followed ( or preceded ) by three dots.

`"For example,"` `{1, 2, 3 . . .}` is the set of natural numbers,

` \ \ \ \ \ \ \ \ \ \ \ {1, 3, 5, 7, . . .} ` is the set of odd natural numbers,

` \ \ \ \ \ \ \ \ \ \ \ {. . .,–3, –2, –1, 0,1, 2 ,3, . . .}` is the set of integers. All these sets are infinite.

`\color{green} ✍️ \color{green} \mathbf(KEY \ POINTS)`

• All infinite sets cannot be described in the roster form.

• For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

`\color{green} ✍️ \color{green} \mathbf( "Cardinality of a finite set" )`

`"Cardinality"` of a finite Set is defined as total number of elements in a set.

Q 3017267189

State which of the following sets are finite or infinite :form.

`(i)\ \ {x : x in N` and `(x . 1) (x .2) = 0}`

`(ii)\ \ {x : x in N` and `x^2 = 4}`

`(iii)\ \ {x : x in N` and `2x .1 = 0}`

`(iv)\ \ {x : x in N` and `x` is prime}

`(v) \ \{x : x in N` and `x` is odd}

`(i)\ \ {x : x in N` and `(x . 1) (x .2) = 0}`

`(ii)\ \ {x : x in N` and `x^2 = 4}`

`(iii)\ \ {x : x in N` and `2x .1 = 0}`

`(iv)\ \ {x : x in N` and `x` is prime}

`(v) \ \{x : x in N` and `x` is odd}

(i) Given set `= {1, 2}.` Hence, it is finite.

(ii) Given set `= {2}.` Hence, it is finite.

(iii) Given set `= phi`. Hence, it is finite.

(iv) The given set is the set of all prime numbers and since set of prime

numbers is infinite. Hence the given set is infinite

(v) Since there are infinite number of odd numbers, hence, the given set is

infinite.

Q 1914034850

Which of the following sets arc finite or infinite:

(i) The set of months of `2` year

(ii) `{1, 2, 3, ... }`

(iii) `{1, 2, 3, ... , 99, 100}`

(iv) The set of positive integers greater than `100`.

(v) The set of prime numbers less than `99`.

Class 11 Exercise 1.2 Q.No. 2

(i) The set of months of `2` year

(ii) `{1, 2, 3, ... }`

(iii) `{1, 2, 3, ... , 99, 100}`

(iv) The set of positive integers greater than `100`.

(v) The set of prime numbers less than `99`.

Class 11 Exercise 1.2 Q.No. 2

(i) Finite

(ii) Infinite

(iii) Finite

(iv) Infinite

(v) Finite.

Q 1984034857

State whether each of the following set is finite or infinite:

(i) The set of lines which are parallel to the `x`-axis.

(ii) The set of letters in the English alphabet.

(iii) The set of numbers which are multiple of `5`.

(iv) The set of animals living on the earth.

(v) The set of circles passing through the origin `(0, 0)`.

Class 11 Exercise 1.2 Q.No. 3

(i) The set of lines which are parallel to the `x`-axis.

(ii) The set of letters in the English alphabet.

(iii) The set of numbers which are multiple of `5`.

(iv) The set of animals living on the earth.

(v) The set of circles passing through the origin `(0, 0)`.

Class 11 Exercise 1.2 Q.No. 3

(i) Infinite

(ii) Finite

(iii) Infinite

(iv) Finite

(v) Infinite

Given two sets `A` and `B,` if every element of `A` is also an element of `B` and if every element of `B` is also an element of `A,` then the sets `A` and `B` are said to be equal. Clearly, the two sets have exactly the same elements.

`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"`

Two sets `A` and `B` are said to be equal if they have exactly the same elements and we write `A = B.` Otherwise, the sets are said to be unequal and we write `A ≠ B`.

`"We consider the following examples :"`

(i) Let `A = {1, 2, 3, 4}` and `B = {3, 1, 4, 2}`. Then `A = B`.

(ii) Let `A` be the set of prime numbers less than `6` and `P` the set of prime factors of `30`. Then `A` and `P` are equal, since `2, 3` and `5` are the only prime factors of `30` and also these are less than `6`.

`\color{green} ✍️ \color{green} \mathbf(KEY \ POINT)`

A set does not change if one or more elements of the set are repeated.

For example, the sets `A = {1, 2, 3}` and `B = {2, 2, 1, 3, 3}` are equal, since each element of `A` is in `B` and vice-versa .

`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"`

Two sets `A` and `B` are said to be equal if they have exactly the same elements and we write `A = B.` Otherwise, the sets are said to be unequal and we write `A ≠ B`.

`"We consider the following examples :"`

(i) Let `A = {1, 2, 3, 4}` and `B = {3, 1, 4, 2}`. Then `A = B`.

(ii) Let `A` be the set of prime numbers less than `6` and `P` the set of prime factors of `30`. Then `A` and `P` are equal, since `2, 3` and `5` are the only prime factors of `30` and also these are less than `6`.

`\color{green} ✍️ \color{green} \mathbf(KEY \ POINT)`

A set does not change if one or more elements of the set are repeated.

For example, the sets `A = {1, 2, 3}` and `B = {2, 2, 1, 3, 3}` are equal, since each element of `A` is in `B` and vice-versa .

Q 3037367282

Find the pairs of equal sets, if any, give reasons:

`A = {0},`

`B = {x : x > 15 and x < 5},`

`C = {x : x – 5 = 0 },`

`D = {x: x^2 = 25}`,

`E = {x : x` is an integral positive root of the equation `x^2 – 2x –15 = 0}.`

`A = {0},`

`B = {x : x > 15 and x < 5},`

`C = {x : x – 5 = 0 },`

`D = {x: x^2 = 25}`,

`E = {x : x` is an integral positive root of the equation `x^2 – 2x –15 = 0}.`

Since `0 ∈ A` and `0` does not belong to any of the sets `B, C, D` and `E`, it

follows that, `A ≠ B, A ≠ C, A ≠ D, A ≠ E.`

Since `B = phi` but none of the other sets are empty. Therefore `B ≠ C, B ≠ D`

and `B ≠ E`. Also `C = {5}` but `–5 ∈ D,` hence `C ≠ D.`

Since `E = {5}, C = E`. Further, `D = {–5, 5}` and `E = {5}`, we find that, `D ≠ E.`

Thus, the only pair of equal sets is `C` and `E.`

Q 3067367285

Which of the following pairs of sets are equal? Justify your answer.

`(i) X`, the set of letters in `“"ALLOY"”` and B, the set of letters in “LOYAL”.

`(ii) A = {n : n ∈ Z` and `n^2 ≤ 4}` and `B = {x : x ∈ R` and `x^2 – 3x + 2 = 0}.`

`(i) X`, the set of letters in `“"ALLOY"”` and B, the set of letters in “LOYAL”.

`(ii) A = {n : n ∈ Z` and `n^2 ≤ 4}` and `B = {x : x ∈ R` and `x^2 – 3x + 2 = 0}.`

(i) We have, `X = {A, L, L, O, Y}, B = {L, O, Y, A, L}.` Then X and B are

equal sets as repetition of elements in a set do not change a set. Thus,

`X = {A, L, O, Y} = B`

`(ii) A = {–2, –1, 0, 1, 2}, B = {1, 2}.` Since `0 ∈ A` and `0 ∉ B, A` and `B` are not equal sets.

Q 3037578482

Show that the set of letters needed to spell CATARACT ” and the set of letters needed to spell “ TRACT” are equal.

Let X be the set of letters in “CATARACT”. Then

`X = { C, A, T, R }`

Let Y be the set of letters in “ TRACT”. Then

`Y = { T, R, A, C, T } = { T, R, A, C }`

Since every element in X is in Y and every element in Y is in X. It follows that X = Y.

Q 1904134958

In the following, state whether `A= B` or not:

(i) `A = { a , b , c , d } , B = { d , c , b , a }`

(ii) `A= {4, 8, 12, 16} , B = {8, 4, 16, 18}`

(iii) `A= {2, 4, 6, 8, 10}`

`B = {x : x` is positive even integer and `x le 10}`

(iv} `A = {x : x` is a multiple of `10}`

`B = { 10, 15, 20, 25, 30, ... }`

Class 11 Exercise 1.2 Q.No. 4

(i) `A = { a , b , c , d } , B = { d , c , b , a }`

(ii) `A= {4, 8, 12, 16} , B = {8, 4, 16, 18}`

(iii) `A= {2, 4, 6, 8, 10}`

`B = {x : x` is positive even integer and `x le 10}`

(iv} `A = {x : x` is a multiple of `10}`

`B = { 10, 15, 20, 25, 30, ... }`

Class 11 Exercise 1.2 Q.No. 4

(i) Yes, `A = B`

(ii) No, `A ne B`

(iii) Yes, `A = B`

(iv) No, `A ne B`.

Q 1924245151

Are the following pairs of sets equal? Gives reasons.

(i) `A = {2, 3}, B = {x : x` is solution of `x^2 + 5x + 6 = 0}`

(ii) `A= { x : x` is a letter in the word `text ('FOLLOW'})`

`B = {y : y` is a letter in the word `text ('WOLF'})`

Class 11 Exercise 1.2 Q.No. 5

(i) `A = {2, 3}, B = {x : x` is solution of `x^2 + 5x + 6 = 0}`

(ii) `A= { x : x` is a letter in the word `text ('FOLLOW'})`

`B = {y : y` is a letter in the word `text ('WOLF'})`

Class 11 Exercise 1.2 Q.No. 5

(i) No, `A= {2, 3}`

But solution of `x^2 + 5x + 6 = 0`

`=> x^2 + 3x + 2x + 6 = 0`

`=> x ( x + 3 ) + 2 (x +3) =0`

`=> ( x + 3 ) ( x + 2 ) = 0`

`=> x + 3 = 0` or `x + 2 = 0`

`=> x = -3` or `x = - 2`

`=> B= {-2, -3}`

Hence,`A ne B`

(ii) Yes, `A= { F, O, L, W}`

and `B = {W,O, L, F}`

Hence, `A=B`

Q 1974245156

From the sets given below, select equal sets:

`A= {2, 4, 8, 12}`,

`B = {1, 2 , 3 , 4 }`,

`C= {4, 8, 12, 14}`,

`D = {3, 1, 4, 2}`,

`E = {-1, 1 }, F = {0, a}` ,

`G = { 1 , - 1 } , H = { 0 , 1 }`

Class 11 Exercise 1.2 Q.No. 6

`A= {2, 4, 8, 12}`,

`B = {1, 2 , 3 , 4 }`,

`C= {4, 8, 12, 14}`,

`D = {3, 1, 4, 2}`,

`E = {-1, 1 }, F = {0, a}` ,

`G = { 1 , - 1 } , H = { 0 , 1 }`

Class 11 Exercise 1.2 Q.No. 6

(i) `B = D `,

(ii) `E = G`