Which of the following given below is null set?

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

(ii). Set of even numbers which are prime

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

(iv). {x: x is a point common to any two parallel lines}

(i) Null Set

A set of odd natural numbers which are divisible by 2 is a null set as none of the odd numbers is divisible by 2.

(ii) Not a Null set

A set of even prime numbers is not null set as there is number 2 which is prime and divisible by 2.

(iii) Null set

{x: x is a natural number, x<5 and x>7} is a null set as any number cannot be less than 5 and greater than 7.

(iv) Null set

{x: x is a point common to any two parallel lines} is a null set as parallel lines do not intersect. Therefore, there is no common point.

State whether the following sets are infinite or finite:

(i). A set of months of a year.

(ii). {1, 2, 3 ….}

(iii). {1, 2, 3…99, 100}

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

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

(i). Finite set as set of months of a year has 12 elements.

(ii) Infinite set as set {1, 2, 3 ….} has infinite numbers of elements

(iii) Finite set as set {1, 2, 3…99, 100} has elements from 1 to 100.

(iv)Infinite set as set of positive integers which are greater than 100 has infinite elements as there are infinite such elements

(v) Finite set as set of prime numbers which are less than 99 has finite numbers.

State whether the following sets are infinite or finite:

(i). The set of lines parallel to the x – axis.

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

(iii). The set of numbers multiple of 5.

(iv). The set of humans living on Earth.

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

(i) infinite set

(ii) finite set as set of letters in the alphabet has a finite element that is 26 elements.

(iii) infinite set as set of numbers which are multiple of 5 has infinite elements.

(iv) finite set as set of animals living on the earth has a finite number of elements.

(v) The set of circles passing through the origin (0, 0) has infinite elements as number of circles can pass through the origin. Therefore, it is an infinite set.

In the following set given below, state whether A = B or not:

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

B = {d, c, b, a}

(ii). A = {4, 8, 12, 16}

B = {16, 8, 12, 18}

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

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

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

B = {10, 15, 20, 25, 30 …}

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

B = {d, c, b, a}

Both the sets have same elements but the order is different. Therefore, A = B

(ii). A = {4, 8, 12, 16}

B = {16, 8, 12, 18}

We see that 18 ∈ A but 18 ∉ B. Therefore A ≠ B

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

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

= {2, 4, 6, 8, 10}

Therefore, A = B

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

B = {10, 15, 20, 25, 30 …}

We see that 15 ∈ B but 15 ∉ A.

Therefore A ≠ B

In the following set given below, is the pair of sets equal?

(i). A = {2, 3}

B = {x: x is solution of x

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

B = {x: x is a letter in the word WOLF}

(i) A = {2, 3}

B = {x: x is solution of x

The equation given x

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

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

x = –2 or x = –3

Therefore, A = {2, 3} and B = {–2, –3}

Therefore, A ≠ B

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

= {F, O, L, W}

B = {x: x is a letter in the word WOLF}

= {W, O, L, F}

Both the sets have same elements but the order is different.

Therefore, A = B

From the following sets, 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}

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}

Lets start checking one by one

8 ∈ A, 8 ∉ B, 8 ∉ D, 8 ∉ E, 8 ∉ F, 8 ∉ G and 8 ∉ H

Therefore, A ≠ B, A ≠ D, A ≠ E, A ≠ F, A ≠ G and A ≠ H

Also,

2 ∈ A and 2 ∉ C

Therefore, A ≠ C

Also, 3 ∈ B, 3 ∉ C, 3 ∉ E, 3 ∉ F, 3 ∉ G and 3 ∉ H

Therefore, B ≠ C, B ≠ E, B ≠ F, B ≠ G, B ≠ H

But we can see that B=D

12 ∈ C, 12 ∉ D, 12 ∉ E, 12 ∉ F, 12 ∉ G, 12 ∉ H

Therefore, C ≠ D, C ≠ E, C ≠ F, C ≠ G and C ≠ H

Also,

4 ∈ D, 4 ∉ E, 4 ∉ F, 4 ∉ G, 4 ∉ H

Therefore, D ≠ E, D ≠ F, D ≠ G, D ≠ H

Similarly, E ≠ F, E ≠ H, F ≠ G, F ≠ H and G ≠ H

But we can that E=G

The order in which elements of the set are listed is not significant.

Therefore, B = D and E = G

Therefore, they are equal.

Fill in the blanks properly using ⊂ and ⊂?.

(i). {2, 3, 4} ____ {1, 2, 3, 4, 5}

(ii). {a, b, c} ____ {b, c, d}

(iii). {x: x is a pupil of Class XI of the school} ____ {x: x is students of the school}

(iv). {x: x is a circle in the plane} ____ {x: x is a circle in the same plane with radius 1 unit}

(v). {x: x is an equilateral triangle in a plane} ____ {x: x is a rectangle in the same plane}

(vi). {x: x is an equilateral triangle in a plane} ____ {x: x is a triangle in the plane}

(vii). {x: x is an odd natural number} ____ {x: x is an integer}

(i). {2, 3, 4} ⊂ {1,2, 3, 4, 5}

(ii). {a, b, c} ⊂? {b, c, d}

(iii). {x: x is a pupil of Class XI of the school} ⊂ {x: x is students of the school}

(iv). {x: x is a circle in the plane} ⊂? {x: x is a circle in the same plane with radius 1 unit}

(v). {x: x is an equilateral triangle in a plane} ⊂? {x: x is a rectangle in the same plane}

(vi). {x: x is an equilateral triangle in a plane} ⊂? {x: x is a triangle in the same plane}

(vii). {x: x is an even natural number} ⊂ {x: x is an integer}

State whether the given statements are true or false:

(i) {a,

(ii) {a,

(iii) {1, 2, 3} ⊂ {1, 3

(iv) {a

(v) {a

(vi) {x

(i). False

As {a,

(ii). True

Since, a, e and i are the three vowels of the English alphabet.

(iii). False

Since, 2 ∈ {1, 2, 3} but 2 ∉ {1, 3, 5}

(iv). True

Since, Element a of {a} is also present in {a, b, c}.

(v). False

Since, Element {a} is not present in {a, b, c}.

(vi). True

Since, {x: x is an even natural number less than 6} = {2, 4}

{x: x is a natural no. which can divide 36} = {1, 2, 3, 4, 6, 9, 12, 18, 36}

Each element of {2, 4} are present in {1, 2, 3, 4, 6, 9, 12, 18, 36}

Let A = {1, 2, {3, 4}, 5}. According to the given set which of the given statements are false? Explain why.

(i). {3, 4} ⊂ A

(ii). {3, 4} ∈ A

(iii). {{3, 4}} ⊂ A

(iv). 1 ∈ A

(v). 1 ⊂ A

(vi). {1, 2, 5} ⊂ A

(vii). {1, 2, 5} ∈ A

(viii). {1, 2, 3} ⊂ A

(ix). Ø ∈ A

(x). Ø ⊂ A

(xi). {Ø} ⊂ A

Given:

A = {1, 2, {3, 4}, 15}

(i). The Statement {3, 4} ⊂ A is False

Since, 3 ∈ {3, 4}

But, 3 ∉ A

(ii). The Statement {3, 4} ∈ A is True

As, {3, 4} is an element of A

(iii). The Statement {{3, 4}} ⊂ A is True

As, {3, 4} ∈ {{3, 4}} and {3, 4} ∈ X

(iv). The Statement 1 ∈ A is True

As clearly 1 is an element of A

(v). The Statement 1 ⊂ A is False

Since, an element of a set can never be subset of itself.

(vi). The Statement {1, 2, 5} ⊂ A is True

Since, each element of {1, 2, 5} is present in X

(vii). The Statement {1, 2, 5} ∈ A is False

Since, {1, 2, 3} is not an element of A.

(viii). The Statement {1, 2, 3} ⊂ A is False

Since, 3 ∈ {1, 2, 3}

But, 3 ∉ A

(ix). The Statement Ø ∈ A is False

Since, A does not contain element Ø

(x). The Statement Ø ⊂ A is True

Since, Ø is subset of every set.

(xi). The Statement {Ø} ⊂ A is False

Since, Ø ∈ {Ø} and Ø does not belong to A

Write all the subsets of the given sets:

(i). {a}

(ii). {a, b}

(iii). {1, 2, 3}

(iv). Ø

(i). {a}:

Subsets are as given: Ø and {a}

(ii) {a, b}:

Subsets are as given: Ø, {a}, {b} and {a, b}

(iii). {1, 2, 3}:

Subsets are as given: Ø, {1}, 2}, {3}, {1, 2}, {1, 3}, {2, 3} and {1, 2, 3}

(iv). Ø:

Subsets are as given: Ø

How many elements has P(A), if A = Ø?

If A has m elements that is n(A) = m, then n[P(A)] = 2

If A = Ø, then n(A) = 0

Therefore, n[P(A)] = 2

Therefore, P(A) has one element.

Write the given in the form of intervals:

(i). {x: x∈ R, –4 < x ≤ 6}

(ii). {x: x∈ R, –12 < x < –10}

(iii). {x: x∈ R, 0 ≤ x < 7}

(iv). {x: x∈ R, 3 ≤ x ≤ 4}

(i). {x: x∈ R, –4 < x ≤ 6} = (-4, 6]

(ii). x: x∈ R, –12 < x < –10} = (-12, -10)

(iii). {x: x∈ R, 0 ≤ x < 7} = [0, 7)

(iv). {x: x∈ R, 3 ≤ y ≤ 4} = [3, 4]

Write the given intervals in the form of set – builder:

(i). (–3, 0)

(ii). [6, 12]

(iii). (6, 12]

(iv). [–23, 5)

(i). (–3, 0) = {x: x∈ R, -3 < y < 0}

(ii). [6, 12] = {x: x∈ R, 6 ≤ y ≤ 12}

(iii). (6, 12] = {x: x∈ R, 6 < y ≤ 12}

(iv). [–23, 5) = {x: x∈ R, –23 ≤ x < 5}

What universal set/ sets would you propose for the given sets?

(i). The set of right triangles

(ii). The set of isosceles triangles

(i)For the set of right triangles, the universal set can be the set of triangles or the set of polygons.

(ii) For the set of isosceles triangles, the universal set can be the set of triangles or the set of polygons or the set of two-dimensional figures.

A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}

Which of the given sets can be considered as the universal set for the given sets A, B and C?

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

(ii). Ø

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

(iv). {1, 2, 3, 4, 5, 6, 7, 8}

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

A ⊂ {0, 1, 2, 3, 4, 5, 6}

B ⊂ {0, 1, 2, 3, 4, 5, 6}

C ⊂? {0, 1, 2, 3, 4, 5, 6}

Therefore, the set {0, 1, 2, 3, 4, 5, 6} cannot be the universal set for the sets A, B and C.

(ii). Ø

X ⊂? Ø

Y ⊂? Ø

Z ⊂? Ø

Therefore, the set Ø cannot be the universal set for the sets A, B and C.

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

A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Hence ,set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for the sets A, B and C.

(iv). {1, 2, 3, 4, 5, 6, 7, 8}

A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

C ⊂? {1, 2, 3, 4, 5, 6, 7, 8}

Hence ,set {1, 2, 3, 4, 5, 6, 7, 8} cannot be the universal set for the sets A, B and C.

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**Notes**- Introduction
- Methods of representing a set
- Types of sets
- Subset
- Subset
- Subset of set of the real numbers
- Interval as subset of R Real Number
- Power Set
- Universal Set
- Venn diagram
- Operation on Sets
- Cardinality of Sets
- Set Theory Symbols

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