TOPICS

Miscellaneous

Sets

**Question-1 :-** Decide, among the following sets, which sets are subsets of one and another: A = { x : x ∈ R and x satisfy x² – 8x + 12 = 0 }, B = { 2, 4, 6 }, C = { 2, 4, 6, 8, . . . }, D = { 6 }.

A = {x: x ∈ R and x satisfies x² – 8x + 12 = 0} x² – 8x + 12 = 0 x² – 6x - 2x + 12 = 0 x(x - 6) - 2(x - 6) = 0 (x - 6) (x - 2) = 0 6 and 2 are the only solutions of x² – 8x + 12 = 0. ∴ A = {2, 6} B = {2, 4, 6}, C = {2, 4, 6, 8...}, D = {6} ∴ D ⊂ A ⊂ B ⊂ C Hence, A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B, D ⊂ C

**Question-2 :-** In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(i) If x ∈ A and A ∈ B , then x ∈ B

(ii) If A ⊂ B and B ∈ C , then A ∈ C

(iii) If A ⊂ B and B ⊂ C , then A ⊂ C

(iv) If A ⊄ B and B ⊄ C , then A ⊄ C

(v) If x ∈ A and A ⊄ B , then x ∈ B

(vi) If A ⊂ B and x ∉ B , then x ∉ A

(i) Statement is False. e.g., A = {a, b} and B = {a, {a, b}, {c}} b ∈ {a, b} and {a, b} ∈ {a, {a, b}, {c}} but b ∉ {a, {a, b}, {c}}

(ii) Statement is False. e.g., A = {1}, B = {0, 1}, C = {2, {0, 1}, {3}} Here, A ⊂ B and B ∈ C but A ∉ C.

(iii) Statement is True. e.g., A = {1}, B = {0, 1}, C = {2, 0, 1, 3} Here, A ⊂ B and B ⊂ C then A ⊂ C.

(iv) Statement is False. e.g., A = {1, 2}, B = {3, 4}, C = {1, 2, 3, 5, 6} Here, A ⊄ B and B ⊄ C then A ⊂ C.

(v) Statement is False. e.g., A = {3, 5, 7} and B = {3, 4, 6} Here, 5 ∈ A and A ⊄ B then, 5 ∉ B

(vi) Statement is True. e.g., A = {1, 2} B = {1, 2, 3} Here, A ⊂ B and 4 ∉ B. To show: 4 ∉ A If possible, suppose 4 ∈ A. Then, 4 ∈ B, which is a contradiction as x ∉ B ∴x ∉ A

**Question-3 :-** Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.

Let, A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. To show : B = C x ∈ (A ∪ B) [B ⊂ (A ∪ B)] x ∈ (A ∪ C) [A ∪ B = A ∪ C] x ∈ A or x ∈ C Case I x ∈ A . Also, x∈B x ∈ A ∩ B x ∈ A ∩ C [A ∩ B = A ∩ C] x ∈ A and x ∈ C ∴ x ∈ C ∴ B ⊂ C Similarly, we can show that C ⊂ B. ∴ B = C

**Question-4 :-** Show that the following four conditions are equivalent:

(i) A ⊂ B, (ii) A – B = Φ, (iii) A ∪ B = B, (iv) A ∩ B = A

First, we have to show that (i) ⇔ (ii). Let A ⊂ B To show: A – B ≠ Φ If possible, suppose A – B ≠ Φ This means that there exists x ∈ A, x ≠ B, which is not possible as A ⊂ B ∴ A – B = Φ ∴ A ⊂ B ⇒ A – B = Φ Let A – B = Φ To show: A ⊂ B Let x ∈ A Clearly, x ∈ B because if x ∉ B, then A – B ≠ Φ ∴ A – B = Φ ⇒ A ⊂ B Hence, (i) ⇔ (ii) Now, we have to show that (i) ⇔ (iii) Let A ⊂ B To show: A ∪ B = B Clearly: B ⊂ A ∪ B Let x ∈ A or x ∈ B Case : x ∈ A ∴ x ∈ B [A ⊂ B] ∴ A ∪ B ⊂ B Case II : x ∈ B Then, A ∪ B = B Converrsely, let A ∪ B = B Let x ∈ A ∴ x ∈ A ∪ B [A ⊂ A ∪ B] ∴ x ∈ B [A ∪ B] ∴ A ⊂ B Hence, (i) ⇔ (iii). Now, we have to show that (i) ⇔ (iv). Let A ⊂ B Clearly A ∩ B ⊂ A Let x ∈ A We have to show that x ∈ A ∩ B As A ⊂ B, x ∈ A ∴ x ∈ A ∩ B ∴ A ⊂ A ∩ B Hence, A = A ∩ B Conversaly, suppose A ∩ B = A Let x ∈ A ∴ x ∈ A ∩ B ∴ x ∈ A and x ∈ B ∴ x ∈ B ∴ A ⊂ B Hence, (i) ⇔ (iv).

**Question-5 :-** Show that if A ⊂ B, then C – B ⊂ C – A.

Given that A ⊂ B To show : C – B ⊂ C – A Let x ∈ C – B ⇒ x ∈ C and x ∉ B ⇒ x ∈ C and x ∉ A [A ⊂ B] ⇒ x ∈ C – A ∴ C – B ⊂ C – A

**Question-6 :-** Assume that P (A) = P (B). Show that A = B.

Given that P(A) = P(B) To show: A = B Let x ∈ A A ∈ P(A) = P(B) ∴ x ∈ C, for some C ∈ P(B). Now, C ⊂ B ∴ x ∈ B ∴ A ⊂ B Similarly, B ⊂ A ∴ A = B

**Question-7 :-** Is it true that for any sets A and B, P (A) ∪ P (B) = P (A ∪ B)? Justify your answer.

False Let A = {1, 2} and B = {2, 3} ∴ A ∪ B = {1, 2, 3} P(A) = {Φ, {1}, {2}, {1, 2}} P(B) = {Φ, {2}, {3}, {2, 3}} P(A ∪ B) = {Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1,2, 3}} P(A) ∪ P(B) = {Φ, {1}, {2}, {1, 2}, {3}, {2, 3} ∴ P(A) ∪ P(B) ≠ P(A ∪ B)

**Question-8 :-** Show that for any sets A and B, A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B).

(i) To show: A = (A ∩ B) ∪ (A – B) Let x ∈ A We have to show that x ∈ (A ∩ B) ∪ (A – B) Case I x ∈ A ∩ B Then, x ∈ (A ∩ B) ⊂ (A ∪ B) ∪ (A – B) Case II x ∉ A ∩ B ⇒ x ∉ A or x ∉ B ∴ x ∉ B [x ∉ A] ∴ x ∉ A – B ⊂ (A ∪ B) ∪ (A – B) ∴ A ⊂ (A ∩ B) ∪ (A – B).............. (1). It is clear that A ∩ B ⊂ A and (A – B) ⊂ A ∴ (A ∩ B) ∪ (A – B) ⊂ A...............(2) From (1) and (2), we obtain A = (A ∩ B) ∪ (A – B)

(ii) To prove: A ∪ (B – A) ⊂ A ∪ B Let x ∈ A ∪ (B – A) ⇒ x ∈ A or x ∈ (B – A) ⇒ x ∈ A or (x ∈ B and x ∉ A) ⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∉ A) ⇒ x ∈ (A ∪ B) ∴ A ∪ (B – A) ⊂ (A ∪ B)..............(3) Next, we show that (A ∪ B) ⊂ A ∪ (B – A). Let y ∈ A ∪ B ⇒ y ∈ A or y ∈ B ⇒ (y ∈ A or y ∈ B) and (y ∈ A or y ∉ A) ⇒ y ∈ A or (y ∈ B and y ∉ A) ⇒ y ∈ A ∪ (B – A) ∴ A ∪ B ⊂ A ∪ (B – A)............... (4) Hence, from (3) and (4), we obtain A ∪ (B – A) = A ∪ B.

**Question-9 :-** Using properties of sets show that

(i) A ∪ (A ∩ B) = A, (ii) A ∩ (A ∪ B) = A.

(i) To show: A ∪ (A ∩ B) = A We know that A ⊂ A A ∩ B ⊂ A ∴ A ∪ (A ∩ B) ⊂ A................. (1) Also, A ⊂ A ∪ (A ∩ B)........... (2) ∴ From (1) and (2), we obtain A ∪ (A ∩ B) = A

(ii) To show: A ∩ (A ∪ B) = A A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B) = A ∪ (A ∩ B) = A {from (1)}

**Question-10 :-** Show that A ∩ B = A ∩ C need not imply B = C.

Let A = {a, b}, B = {a, c, d}, and C = {a, e, f} Accordingly, A ∩ B = {a} and A ∩ C = {a} Here, A ∩ B = A ∩ C = {a} However, B ≠ C [d ∈ B and d ∉ C]

**Question-11 :-** Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B.

(Hints A = A ∩ ( A ∪ X ) , B = B ∩ ( B ∪ X ) and use Distributive law )

Given that A and B be two sets such that A ∩ X = B ∩ X = f and A ∪ X = B ∪ X for some set X. To show: A = B It can be seen that A = A ∩ (A ∪ X) = A ∩ (B ∪ X) [A ∪ X = B ∪ X] = (A ∩ B) ∪ (A ∩ X) [Distributive law] = (A ∩ B) ∪ Φ [A ∩ X = Φ] = A ∩ B........................(1) Now, B = B ∩ (B ∪ X) = B ∩ (A ∪ X) [A ∪ X = B ∪ X] = (B ∩ A) ∪ (B ∩ X) [Distributive law] = (B ∩ A) ∪ Φ [B ∩ X = Φ] = B ∩ A = A ∩ B....................... (2) Hence, from (1) and (2), we obtain A = B.

**Question-12 :-** Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = Φ.

Let A = {1, 2}, B = {2, 3}, and C = {1, 3}. Accordingly, A ∩ B = {2}, B ∩ C = {3}, and A ∩ C = {1}. ∴ A ∩ B, B ∩ C, and A ∩ C are non-empty. However, A ∩ B ∩ C = Φ

**Question-13 :-** In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

Let U be the set of all students who took part in the survey. Let T be the set of students taking tea. Let C be the set of students taking coffee. Accordingly, n(U) = 600, n(T) = 150, n(C) = 225, n(T ∩ C) = 100 To find: Number of student taking neither tea nor coffee i.e., we have to find n(T' ∩ C') = n(T ∪ C)' = n(U) – n(T ∪ C) = n(U) – [n(T) + n(C) – n(T ∩ C)] = 600 – [150 + 225 – 100] = 600 – 275 = 325 Hence, 325 students were taking neither tea nor coffee.

**Question-14 :-** In a group of students 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?

Let U be the set of all students in the group. Let E be the set of all students who know English. Let H be the set of all students who know Hindi. ∴ H ∪ E = U Accordingly, n(H) = 100 and n(E) = 50 n(H ∩ E) = 25 n(U) = n(H) + n(E) - n(H ∩ E) n(U) = 100 + 50 - 25 n(U) = 150 - 25 n(U) = 125 Hence, there are 125 students in the group.

**Question-15 :-** In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I,11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:

(i) the number of people who read at least one of the newspapers.

(ii) the number of people who read exactly one newspaper.

Let U be the set of people who took part in the survey. Let A be the set of people who read newspaper H. Let B be the set of people who read newspaper T. Let C be the set of people who read newspaper I. Accordingly, n(A) = 25, n(B) = 26, and n(C) = 26 n(A ∩ C) = 9, n(A ∩ B) = 11, and n(B ∩ C) = 8 n(A ∩ B ∩ C) = 3 (i) Accordingly, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C) = 25 + 26 + 26 – 11 – 8 – 9 + 3 = 80 - 28 = 52 Hence, 52 people read at least one of the newspapers.

(ii) Let a be the number of people who read newspapers H and T only. Let b denote the number of people who read newspapers I and H only. Let c denote the number of people who read newspapers T and I only. Let d denote the number of people who read all three newspapers. Accordingly, d = n(A ∩ B ∩ C) = 3. Now, n(A ∩ B) = a + d n(B ∩ C) = c + d n(C ∩ A) = b + d ∴ a + d + c + d + b + d = 11 + 8 + 9 = 28 ⇒ a + b + c + d = 28 – 2d = 28 – 6 = 22 Hence, (52 – 22) = 30 people read exactly one newspaper.

**Question-16 :-** In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

Let A be the set of people who like product A. Let B be the set of people who like product B. Let C be the set of people who like product C. Accordingly, n(A) = 21, n(B) = 26, n(C) = 29, n(A ∩ B) = 14, n(C ∩ A) = 12, n(B ∩ C) = 14, n(A ∩ B ∩ C) = 8 n(C - A - B) = {n(C) - n(C ∩ A) - n(B ∩ C) + n(A ∩ B ∩ C)} = {29 - 12 - 14 + 8} = {37 - 26} = 11 OR The Venn diagram for the given problem can be drawn as It can be seen that number of people who like product C only is {29 – (4 + 8 + 6)} = 11

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