Part A: Union (11 Questions)
Q1). A = {1,2,3}, B = {3,4,5}, find A ∪ B.
Q2). A = {a,b,c}, B = {c,d,e}, C = {e,f}, find A ∪ B ∪ C.
Q3). A = {x | x ≤ 10, x even}, B = {x | x ≤ 10, x multiple of 3}, find A ∪ B.
Q4). U = {1,…,12}, A = {2,4,6,8,10,12}, B = {3,6,9,12}, find A ∪ B.
Q5). A = {x | x is prime ≤ 15}, B = {x | x odd ≤ 15}, find A ∪ B.
Q6). A = {letters in “math”}, B = {letters in “team”}, find A ∪ B.
Q7). A = {1,3,5,7}, B = {2,4,6,8}, C = {5,6,7,8}, find A ∪ (B ∪ C).
Q8). A = {x | x is multiple of 4 ≤ 20}, B = {x | x is multiple of 6 ≤ 20}, find A ∪ B.
Q9). Prove/disprove: (A ∪ B) ∪ C = A ∪ (B ∪ C).
Q10). A = {vowels}, B = {a,e}, C = {i,o,u}, find A ∪ (B ∪ C).
Q11). U = {1,…,15}, A = {multiples of 5}, B = {multiples of 3}, find A ∪ B.
Part B: Intersection (11 Questions)
Q1). A = {1,2,3,4}, B = {3,4,5,6}, find A ∩ B.
Q2). A = {m,a,t,h}, B = {t,e,a,m}, find A ∩ B.
Q3). A = {x | x ≤ 12, x even}, B = {x | x ≤ 12, x prime}, find A ∩ B.
Q4). A = {x | x ≤ 20, x multiple of 4}, B = {x | x ≤ 20, x multiple of 6}, find A ∩ B.
Q5). A = {letters in “algebra”}, B = {letters in “geometry”}, find A ∩ B.
Q6). A = {2,4,6,8}, B = {4,6,8,10}, C = {6,8,10,12}, find A ∩ B ∩ C.
Q7). U = {1,…,15}, A = {multiples of 5}, B = {multiples of 3}, find A ∩ B.
Q8). A = {x | x is odd ≤ 20}, B = {x | x is prime ≤ 20}, find A ∩ B.
Q9). Verify: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Q10). A = {1,…,12}, B = {4,8,12}, C = {3,6,9,12}, find A ∩ (B ∪ C).
Q11). A = {x | x is factor of 36}, B = {x | x is factor of 48}, find A ∩ B.
Part C: Complement (11 Questions)
Q1). U = {1,…,10}, A = {2,4,6,8,10}, find A′.
Q2). U = {a,b,c,d,e}, A = {a,c,e}, find A′.
Q3). U = {1,…,20}, A = {x | x is multiple of 3}, find A′.
Q4). U = {letters of “mathematics”}, A = {vowels in “mathematics”}, find A′.
Q5). U = {1,…,15}, A = {x | x prime ≤ 15}, find A′.
Q6). U = {1,…,12}, A = {2,4,6,8,10,12}, B = {3,6,9,12}, find (A ∪ B)′.
Q7). U = {1,…,12}, A = {2,4,6,8,10,12}, B = {3,6,9,12}, find (A ∩ B)′.
Q8). Verify De Morgan’s Law: (A ∪ B)′ = A′ ∩ B′.
Q9). Verify De Morgan’s Law: (A ∩ B)′ = A′ ∪ B′.
Q10). U = {1,…,20}, A = {multiples of 4}, B = {multiples of 5}, find (A ∪ B)′.
Q11). U = {1,…,12}, A = {1,2,3,4,5,6}, find (A′)′.
Part D: Difference (11 Questions)
Q1). A = {1,2,3,4}, B = {3,4,5,6}, find A − B.
Q2). A = {m,a,t,h}, B = {t,h}, find A − B.
Q3). U = {1,…,10}, A = {2,4,6,8}, B = {6,8,10}, find A − B.
Q4). A = {x | x ≤ 20, x multiple of 4}, B = {x | x ≤ 20, x multiple of 6}, find A − B.
Q5). A = {vowels}, B = {a,e}, find A − B.
Q6). A = {1,2,…,12}, B = {2,4,6,8,10,12}, find B − A.
Q7). A = {multiples of 3 ≤ 30}, B = {multiples of 5 ≤ 30}, find A − B.
Q8). A = {letters in “function”}, B = {letters in “notion”}, find A − B.
Q9). A = {1,…,20}, B = {x | x prime ≤ 20}, find A − B.
Q10). A = {multiples of 6 ≤ 30}, B = {multiples of 4 ≤ 30}, find B − A.
Q11). Verify: A − (B ∪ C) = (A − B) ∩ (A − C).
Part E: Symmetric Difference (11 Questions)
Q1). A = {1,2,3}, B = {3,4,5}, find A Δ B.
Q2). A = {m,a,t,h}, B = {m,a,t,e}, find A Δ B.
Q3). A = {2,4,6}, B = {4,6,8}, find A Δ B.
Q4). A = {x | x prime ≤ 12}, B = {x | x even ≤ 12}, find A Δ B.
Q5). A = {letters in “algebra”}, B = {letters in “geometry”}, find A Δ B.
Q6). A = {multiples of 4 ≤ 20}, B = {multiples of 6 ≤ 20}, find A Δ B.
Q7). U = {1,…,15}, A = {multiples of 3}, B = {multiples of 5}, find A Δ B.
Q8). A = {1,2,3,4}, B = {3,4,5,6}, C = {2,4,6}, find (A Δ B) Δ C.
Q9). Verify: A Δ B = (A − B) ∪ (B − A).
Q10). A = {letters in “triangle”}, B = {letters in “integral”}, find A Δ B.
Q11). A = {prime ≤ 15}, B = {odd ≤ 15}, find A Δ B.
Answer Key — Union
Q1) {1,2,3,4,5}
Q2) {a,b,c,d,e,f}
Q3) {2,4,6,8,10} ∪ {3,6,9} = {2,3,4,6,8,9,10}
Q4) {2,4,6,8,10,12} ∪ {3,6,9,12} = {2,3,4,6,8,9,10,12}
Q5) {2,3,5,7,11,13} ∪ {1,3,5,7,9,11,13,15} = {1,2,3,5,7,9,11,13,15}
Q6) {m,a,t,h} ∪ {t,e,a,m} = {m,a,t,h,e}
Q7) {1,3,5,7} ∪ ({2,4,6,8} ∪ {5,6,7,8}) = {1,2,3,4,5,6,7,8}
Q8) {4,8,12,16,20} ∪ {6,12,18} = {4,6,8,12,16,18,20}
Q9) Always true (associativity law).
Q10) {a,e,i,o,u}
Q11) {multiples of 3 or 5 ≤ 15} = {3,5,6,9,10,12,15}
Answer Key — Intersection
Q1) {3,4}
Q2) {a,t}
Q3) Even ≤ 12 = {2,4,6,8,10,12}, Prime ≤ 12 = {2,3,5,7,11} → Intersection = {2}
Q4) Multiples of 4 ≤ 20 = {4,8,12,16,20}, Multiples of 6 ≤ 20 = {6,12,18} → Intersection = {12}
Q5) {a,g,e} ∩ {g,e,o,m,t,r,y} = {g,e}
Q6) {2,4,6,8} ∩ {4,6,8,10} ∩ {6,8,10,12} = {6,8}
Q7) Multiples of 5 ≤ 15 = {5,10,15}, Multiples of 3 ≤ 15 = {3,6,9,12,15} → Intersection = {15}
Q8) Odd ≤ 20 = {1,3,5,7,9,11,13,15,17,19}, Prime ≤ 20 = {2,3,5,7,11,13,17,19} → Intersection = {3,5,7,11,13,17,19}
Q9) Always true (distributive law).
Q10) {1,…,12} ∩ ({4,8,12} ∪ {3,6,9,12}) = {3,4,6,8,9,12}
Q11) Factors of 36 = {1,2,3,4,6,9,12,18,36}, Factors of 48 = {1,2,3,4,6,8,12,16,24,48} → Intersection = {1,2,3,4,6,12}
Answer Key — Complement
Q1) {1,3,5,7,9}
Q2) {b,d}
Q3) U = {1,…,20}, A = {3,6,9,12,15,18} → A′ = {1,2,4,5,7,8,10,11,13,14,16,17,19,20}
Q4) U = {m,a,t,h,e,i,c,s}, A = {a,e,i} → A′ = {m,t,h,c,s}
Q5) U = {1,…,15}, A = {2,3,5,7,11,13} → A′ = {1,4,6,8,9,10,12,14,15}
Q6) (A ∪ B)′ = {1,5,7,11}
Q7) (A ∩ B)′ = {1,2,3,4,5,7,8,9,10,11}
Q8) True by De Morgan’s Law.
Q9) True by De Morgan’s Law.
Q10) Multiples of 4 ≤ 20 = {4,8,12,16,20}, Multiples of 5 ≤ 20 = {5,10,15,20}, Union = {4,5,8,10,12,15,16,20}, Complement = {1,2,3,6,7,9,11,13,14,17,18,19}
Q11) (A′)′ = A = {1,2,3,4,5,6}
Answer Key — Difference
Q1) {1,2}
Q2) {m,a}
Q3) {2,4}
Q4) {4,8,16,20}
Q5) {i,o,u}
Q6) ∅
Q7) Multiples of 3 ≤ 30 = {3,6,9,12,15,18,21,24,27,30}, Multiples of 5 ≤ 30 = {5,10,15,20,25,30}, A − B = {3,6,9,12,18,21,24,27}
Q8) {f,u,c}
Q9) {1,4,6,8,9,10,12,14,15,16,18,20}
Q10) Multiples of 6 ≤ 30 = {6,12,18,24,30}, Multiples of 4 ≤ 30 = {4,8,12,16,20,24,28}, B − A = {4,8,16,20,28}
Q11) Always true (set identity).
Answer Key — Symmetric Difference
Q1) {1,2,4,5}
Q2) {h,e}
Q3) {2,8}
Q4) {2,3,5,7,11} Δ {2,4,6,8,10,12} = {3,4,5,6,7,8,10,11,12}
Q5) {a,l,g,b,r} Δ {g,e,o,m,t,r,y} = {a,l,b,e,o,m,t,y}
Q6) {4,8,12,16,20} Δ {6,12,18} = {4,6,8,16,18,20}
Q7) {3,6,9,12,15} Δ {5,10,15} = {3,5,6,9,10,12}
Q8) (A Δ B) Δ C = {1,5}
Q9) True by definition: A Δ B = (A − B) ∪ (B − A).
Q10) {t,r} Δ {t,r} = ∅ (all common cancel).
Q11) {2,3,5,7,11,13} Δ {1,3,5,7,9,11,13,15} = {1,2,9,15}
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