**Question 1. Which of the following statements are true? Give a reason to support your answer.**

**(i) For any two sets A and B either A B or B A.**

**(ii) Every subset of an infinite set is infinite.**

**(iii) Every subset of a finite set is finite.**

**(iv) Every set has a proper subset.**

**(v) {a, b, a, b, a, b,….} is an infinite set.**

**(vi) {a, b, c} and {1, 2, 3} are equivalent sets.**

**(vii) A set can have infinitely many subsets.**

**Solution:**

(i)FalseIt is not mandatory for any two set A and B to be either A B or B A.

(ii)FalseLet us consider a set, A = {2,3,4}.

It is finite subset with infinite set N of natural numbers.

(iii)TrueA finite set can never have an infinite subset.

Therefore, every subset of a finite set is finite.

(iv)FalseNull set, also known as empty set doesn’t have a proper subset.

(v)FalseA set can never have duplicate entries.

Removing duplications, the set {a, b} becomes a finite set.

(vi)TrueEquivalent sets have the same number of elements. Both the sets have three elements, therefore, are equivalent.

(vii)FalseLet us consider, a set A = {1}

The subsets of this set A can be ϕ and {1} which are finite.

**Question 2. State whether the following statements are true or false:**

**(i) 1 ∈ {1,2,3}**

**(ii) a ⊂ {b,c,a}**

**(iii) {a} ∈ {a,b,c}**

**(iv) {a, b} = {a, a, b, b, a}**

**(v) The set {x: x + 8 = 8} is the null set.**

**Solution:**

(i)True1 is a part of the given set, therefore it belongs to this set {1, 2, 3} .

(ii)FalseSince, a is an element and not a subset of a set {b, c, a}, therefore this statement is false.

(iii)FalseSince, {a} is not an element but a subset of set {b, c, a}.

(iv)TrueA set cannot have duplicate entries. Therefore, removing duplicate entries from RHS, the sets become equivalent.

(v)FalseGiven, x+8 = 8

Solving we get, x = 0

Therefore, the given set becomes a single ton set with the only element being {0}. Where, it is not a null set.

**Question 3. Decide among the following sets, which are subsets of which:**

**A = {x: x satisfies x**^{2} – 8x + 12 = 0}, B = {2,4,6}, C = {2,4,6,8,….}, D = {6}

^{2}– 8x + 12 = 0}, B = {2,4,6}, C = {2,4,6,8,….}, D = {6}

**Solution:**

We have,

A = x

^{2}– 8x + 12=0Solving we get,

⇒ (x–6) (x–2) =0

Solving for x,

⇒ x = 2 or x = 6

Therefore,

A = {2, 6}

Given,

B = {2, 4, 6}

C = {2, 4, 6, 8}

D = {6}

Therefore,

D ⊂ A ⊂ B ⊂ C

**Question 4. Write which of the following statements are true? Justify your answer.**

**(i) The set of all integers is contained in the set of all rational numbers.**

**(ii) The set of all crows is contained in the set of all birds.**

**(iii) The set of all rectangles is contained in the set of all squares.**

**(iv) The set of all rectangle is contained in the set of all squares.**

**(v) The sets P = {a} and B = {{a}} are equal.**

**(vi) The sets A={x: x is a letter of word “LITTLE”} AND, b = {x: x is a letter of the word “TITLE”} are equal.**

**Solution:**

(i)TrueA rational number is a fractional number which is represented by the form p/q where p and q are integers where q is not equal to 0. Substituting, q = 1, we get p=q, which is an integer.

(ii)TrueCrows are also birds, so all the crows are contained in the set of all birds.

(iii)FalseEvery square can be a rectangle, where the length and breadth of the rectangle are same. But, the reverse is not true, that is, not every rectangle cannot be a square.

(iv)FalseEvery square can be a rectangle, where the length and breadth of the rectangle are same. But, the reverse is not true, that is, not every rectangle cannot be a square.

(v)FalseWe have,

P = {a}

B = {{a}}

But, {a} = P

B = {P}

Hence, they are not equivalent.

(vi)TrueWe have,

A = For “LITTLE”

A = {L, I, T, E} = {E, I, L, T}

B = For “TITLE”

B = {T, I, L, E} = {E, I, L, T}

Therefore,

A = B

**Question 5. Which of the following statements are correct? Write a correct form of each of the incorrect statements.**

**(i) a ⊂ {a, b, c}**

**(ii) {a} {a, b, c}**

**(iii) a {{a}, b}**

**(iv) {a} ⊂ {{a}, b}**

**(v) {b, c} ⊂ {a,{b, c}}**

**(vi) {a, b} ⊂ {a,{b, c}}**

**(vii) ϕ {a, b}**

**(viii) ϕ ⊂ {a, b, c}**

**(ix) {x: x + 3 = 3}= ϕ**

**Solution:**

(i)Falsea is not a subset of given set but belongs to the given set.It is just an element.

Correct form is – a ∈ {a, b, c}

(ii)In this {a} is subset of {a, b, c}Correct form is – {a} ⊂ {a, b, c}

(iii)False‘a’ is not the element of the set.

Correct form is – {a} ∈ {{a}, b}

(iv)False{a} is not a subset of given set.

Correct form is – {a} ∈ {{a}, b}

(v){b, c} is not a subset of given set. But it belongs to the given set.Correct form is – {b, c} ∈ {a,{b, c}}

(vi){a, b} is not a subset of given set.Correct form is – {a, b}⊄{a,{b, c}}

(vii)ϕ does not belong to the given set but it is subset.Correct form is – ϕ ⊂ {a, b}

(viii)TrueIt is the correct form. ϕ is subset of every set.

(ix)x + 3 = 3Evaluating, we get,

x = 0 = {0}, which is not ϕ

Correct form is – {x: x + 3 = 3} ≠ ϕ

**Question 6. Let A = {a, b,{c, d}, e}. Which of the following statements are false and why?**

**(i) {c, d} ⊂ A**

**(ii) {c, d} ∈ A**

**(iii) {{c, d}} ⊂ A**

**(iv) a ∈ A**

**(v) a ⊂ A.**

**(vi) {a, b, e} ⊂ A**

**(vii) {a, b, e}∈ A**

**(viii) {a, b, c} ⊂ A**

**(ix) ϕ∈ A**

**(x) {ϕ} ⊂ A**

**Solution:**

(i)False{c, d} is not a subset of A but it belongs to the set A.

Therefore,

{c, d} ∈ A

(ii)True{c, d} ∈ A

(iii)True{c, d} is a subset of A.

(iv)It is true that a belongs to A.

(v)FalseThe element a is not a subset of A but it belongs to the set A.

(vi)True{a, b, e} is a subset of A.

(vii)False{a, b, e} does not belong to A, {a, b, e} ⊂ A this is the correct form.

(viii)False{a, b, c} is not a subset of A

(ix)Falseϕ is a subset of A.

ϕ ⊂ A.

(x)False{ϕ} is not subset of A, ϕ is a subset of A. Therefore, it is false.

### Question **7. Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:**

**(i) 1 ∈ A**

**(ii) {1, 2, 3} ⊂ A**

**(iii) {6, 7, 8} ∈ A**

**(iv) {4, 5} ⊂ A**

**(v) ϕ ∈ A**

**(vi) ϕ ⊂ A**

**Solution:**

(i)False1 is not an element of the set A.

(ii)True{1,2,3} ∈ A. this is correct form.

(iii)True.The set {6, 7, 8} is an element of A, that is {6, 7, 8} ∈ A.

(iv)True{{4, 5}} is a subset of the specified set A={4, 5}.

(v)FalseΦ is a subset of the given set A, not an element of A.

(vi)TrueΦ is a subset of every set, so it is also a subset of the given set A.

**Question 8. Let A = {ϕ, {ϕ}, 1, {1, ϕ}, 2}. Which of the following are true?**

**(i) ϕ ∈ A**

**(ii) {ϕ} ∈ A**

**(iii) {1} ∈ A**

**(iv) {2, ϕ} ⊂ A**

**(v) 2 ⊂ A**

**(vi) {2, {1}} ⊄A**

**(vii) {{2}, {1}} ⊄ A**

**(viii) {ϕ, {ϕ}, {1, ϕ}} ⊂ A**

**(ix) {{ϕ}} ⊂ A**

**Solution:**

(i)TrueΦ is an element of set A, therefore it belongs to set A. Hence, the given statement is true.

(ii)True{Φ} is an element of set A, and not a subset. Hence, the given statement is true.

(iii)FalseThe subset 1 is not an element of A. Hence, the given statement is false

(iv)True{2, Φ} is a subset of the given set A. Hence, the given statement is true.

(v)False2 is not a subset of set A, it is an element of set A. Hence, the given statement is false.

(vi)True{2, {1}} is not a subset of the given set A. Hence, the given statement is true.

(vii)TrueNeither {2} and nor {1} is a subset of set A. Hence, the given statement is true.

(viii)TrueAll three {ϕ, {ϕ}, {1, ϕ}} are subset of set A. Hence, the given statement is true.

(ix)True{{ϕ}} is a subset of set A. Hence, the given statement is true.