1. SETS (1M + 2M + 4M = 7M)
SHORT ANSWER QUESTIONS ( 4 Marks ) :
1. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10} ; find
i) A ∪ B ii) A ∪ C iii) B ∪ C iv) B ∪ D v) A ∪ B ∪ C vi) A ∪ B ∪ D
vi) B ∪ C ∪ D [𝑩𝑶𝑨𝑹𝑫 𝑴𝑶𝑫𝑬𝑳 𝑷𝑨𝑷𝑬𝑹 − 𝟐]
2. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} ,C = {11, 13, 15} and D = {15, 17}; find
i) A ∩ B ii) B ∩ C iii) A ∩ C ∩ D iv) A ∩ C v) B ∩ D vi) A ∩ (B ∪ C)
vii) A ∩ D viii) A ∩ (B ∪ D) ix) (A ∩ B) ∩ (B ∪ C) x) (A ∪ D) ∩ (B ∪ C)
3. If A = {x: x is a natural number}, B = {x: x is an even natural number} ,
C = {x: x is an odd natural number} and D = {x: x is a prime number}; find
i) A ∩ B ii) A ∩ C iii) A ∩ D iv) B ∩ C v) B ∩ D vi) C ∩ D
4. If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}
and D= {5, 10, 15, 20}; find
i) A – B ii) A – C iii) A – D iv) B – A v) C – A vi) D – A
vii) B – C viii) B – D ix) C – B x) D – B xi) C – D xii) D – C
5. If U= {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ} find the complements of the following sets :
i) A = {𝑎, 𝑏, 𝑐 } ii) B= {𝑑, 𝑒, 𝑓, 𝑔}
iii) C= {𝑎, 𝑐, 𝑒, 𝑔} iv) D= {𝑓, 𝑔, ℎ, 𝑎}
6. If U= {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}. find 𝐴′ , 𝐵′ , 𝐴′ ∩ 𝐵′ , A ∪ B and
hence show that (A ∪ B)′ = 𝐴′ ∩ 𝐵′ .
7. If U= {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
i) (A ∪ B)′ = 𝐴′ ∩ 𝐵′ ii) (A ∩ B)′ = 𝐴′ ∪ 𝐵′
8. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}
Find i) 𝐴′ ii) 𝐵′ iii) (A ∪ C)′ iv) (A ∪ B)′ v) (𝐴′ )′ vi) (B – C)′
, 9. Taking the set of natural numbers as the universal set, write down the complement
of the following sets:
i) {𝑥: 𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟} ii) {𝑥: 𝑥 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟}
iii) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3} iv) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟}
v) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 3 𝑎𝑛𝑑 5}
vi) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑒𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒} vii) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑒𝑓𝑒𝑐𝑡 𝑐𝑢𝑏𝑒}
viii) {𝑥: 𝑥 + 5 = 8} ix) {𝑥: 2𝑥 + 5 = 9}
x) {𝑥: 𝑥 ≥ 7} xi) {𝑥: 𝑥 ∈ 𝑁 𝑎𝑛𝑑 2𝑥 + 1 > 10}
10. Draw appropriate venn diagram for each of the following: [𝑩𝑶𝑨𝑹𝑫 𝑴𝑶𝑫𝑬𝑳 𝑷𝑨𝑷𝑬𝑹 − 𝟏]
i)(A ∪ B)′ ii) 𝐴′ ∩ 𝐵′ iii)(A ∩ B)′ iv) 𝐴′ ∪ 𝐵′
11. Show that A ∪ B = A ∩ B implies A = B.
12. 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
13. Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that
B = C.
14. Show that the following four conditions are equivalent:
i) A ⊂ B ii) A – B = ∅ iii) A ∪ B= B iv) A ∩ B=A
15. Show that for any sets A and B, A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) =A ∪ B.
16. Using properties of sets, show that i) A ∪ (A ∩ B) =A ii) A ∩ (A ∪ B) = A.
17. Let A and B be sets. If A ∩ X = B ∩ X = ∅ 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]
SHORT ANSWER QUESTIONS ( 4 Marks ) :
1. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10} ; find
i) A ∪ B ii) A ∪ C iii) B ∪ C iv) B ∪ D v) A ∪ B ∪ C vi) A ∪ B ∪ D
vi) B ∪ C ∪ D [𝑩𝑶𝑨𝑹𝑫 𝑴𝑶𝑫𝑬𝑳 𝑷𝑨𝑷𝑬𝑹 − 𝟐]
2. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} ,C = {11, 13, 15} and D = {15, 17}; find
i) A ∩ B ii) B ∩ C iii) A ∩ C ∩ D iv) A ∩ C v) B ∩ D vi) A ∩ (B ∪ C)
vii) A ∩ D viii) A ∩ (B ∪ D) ix) (A ∩ B) ∩ (B ∪ C) x) (A ∪ D) ∩ (B ∪ C)
3. If A = {x: x is a natural number}, B = {x: x is an even natural number} ,
C = {x: x is an odd natural number} and D = {x: x is a prime number}; find
i) A ∩ B ii) A ∩ C iii) A ∩ D iv) B ∩ C v) B ∩ D vi) C ∩ D
4. If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}
and D= {5, 10, 15, 20}; find
i) A – B ii) A – C iii) A – D iv) B – A v) C – A vi) D – A
vii) B – C viii) B – D ix) C – B x) D – B xi) C – D xii) D – C
5. If U= {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ} find the complements of the following sets :
i) A = {𝑎, 𝑏, 𝑐 } ii) B= {𝑑, 𝑒, 𝑓, 𝑔}
iii) C= {𝑎, 𝑐, 𝑒, 𝑔} iv) D= {𝑓, 𝑔, ℎ, 𝑎}
6. If U= {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}. find 𝐴′ , 𝐵′ , 𝐴′ ∩ 𝐵′ , A ∪ B and
hence show that (A ∪ B)′ = 𝐴′ ∩ 𝐵′ .
7. If U= {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
i) (A ∪ B)′ = 𝐴′ ∩ 𝐵′ ii) (A ∩ B)′ = 𝐴′ ∪ 𝐵′
8. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}
Find i) 𝐴′ ii) 𝐵′ iii) (A ∪ C)′ iv) (A ∪ B)′ v) (𝐴′ )′ vi) (B – C)′
, 9. Taking the set of natural numbers as the universal set, write down the complement
of the following sets:
i) {𝑥: 𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟} ii) {𝑥: 𝑥 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟}
iii) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3} iv) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟}
v) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 3 𝑎𝑛𝑑 5}
vi) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑒𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒} vii) {𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑒𝑓𝑒𝑐𝑡 𝑐𝑢𝑏𝑒}
viii) {𝑥: 𝑥 + 5 = 8} ix) {𝑥: 2𝑥 + 5 = 9}
x) {𝑥: 𝑥 ≥ 7} xi) {𝑥: 𝑥 ∈ 𝑁 𝑎𝑛𝑑 2𝑥 + 1 > 10}
10. Draw appropriate venn diagram for each of the following: [𝑩𝑶𝑨𝑹𝑫 𝑴𝑶𝑫𝑬𝑳 𝑷𝑨𝑷𝑬𝑹 − 𝟏]
i)(A ∪ B)′ ii) 𝐴′ ∩ 𝐵′ iii)(A ∩ B)′ iv) 𝐴′ ∪ 𝐵′
11. Show that A ∪ B = A ∩ B implies A = B.
12. 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
13. Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that
B = C.
14. Show that the following four conditions are equivalent:
i) A ⊂ B ii) A – B = ∅ iii) A ∪ B= B iv) A ∩ B=A
15. Show that for any sets A and B, A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) =A ∪ B.
16. Using properties of sets, show that i) A ∪ (A ∩ B) =A ii) A ∩ (A ∪ B) = A.
17. Let A and B be sets. If A ∩ X = B ∩ X = ∅ 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]
