π Exercise 1.4 - Sets
Union, Intersection, Difference, and Disjoint Sets - Complete Solutions
1
Find the union of each pair of sets
(i) X = {1, 3, 5}, Y = {1, 2, 3}
(ii) A = {a, e, i, o, u}, B = {a, b, c}
(iii) A = {x : x is natural number and multiple of 3}, B = {x : x is natural number less than 6}
(iv) A = {x : 1 < x β€ 6, x β N}, B = {x : 6 < x < 10, x β N}
(v) A = {1, 2, 3}, B = Ο
π‘ Click to View Solution
1 Understanding Union
Union (A βͺ B): All elements that are in A OR in B (or both)
Write each element ONCE, even if it appears in both sets
Write each element ONCE, even if it appears in both sets
2 Solutions
(i) X βͺ Y = {1, 3, 5} βͺ {1, 2, 3}
Elements: 1 (both), 3 (both), 5 (X only), 2 (Y only)
Answer: {1, 2, 3, 5}
Elements: 1 (both), 3 (both), 5 (X only), 2 (Y only)
Answer: {1, 2, 3, 5}
(ii) A βͺ B = {a, e, i, o, u} βͺ {a, b, c}
All vowels plus letters from B
Answer: {a, b, c, e, i, o, u}
All vowels plus letters from B
Answer: {a, b, c, e, i, o, u}
(iii) A = multiples of 3 = {3, 6, 9, 12, ...}
B = natural numbers < 6 = {1, 2, 3, 4, 5}
A βͺ B includes all from both sets
Answer: {1, 2, 3, 4, 5, 6, 9, 12, ...}
B = natural numbers < 6 = {1, 2, 3, 4, 5}
A βͺ B includes all from both sets
Answer: {1, 2, 3, 4, 5, 6, 9, 12, ...}
(iv) A = {2, 3, 4, 5, 6}, B = {7, 8, 9}
No common elements
Answer: {2, 3, 4, 5, 6, 7, 8, 9}
No common elements
Answer: {2, 3, 4, 5, 6, 7, 8, 9}
(v) A βͺ Ο = {1, 2, 3} βͺ Ο
Union with empty set = original set
Answer: {1, 2, 3}
Union with empty set = original set
Answer: {1, 2, 3}
π‘ Key Property: A βͺ Ο = A (Union with empty set always gives original set)
2
Let A = {a, b}, B = {a, b, c}. Is A β B? What is A βͺ B?
π‘ Click to View Solution
1 Check if A β B
Are all elements of A in B?
a β A, is a β B? YES
b β A, is b β B? YES
Yes, A β B β
a β A, is a β B? YES
b β A, is b β B? YES
Yes, A β B β
2 Find A βͺ B
A βͺ B = {a, b} βͺ {a, b, c}
All elements: a, b, c
A βͺ B = {a, b, c} = B
All elements: a, b, c
A βͺ B = {a, b, c} = B
Yes, A β B
A βͺ B = {a, b, c} = B
A βͺ B = {a, b, c} = B
3
If A β B, then what is A βͺ B?
π‘ Click to View Solution
1 Logic
If A β B, then every element of A is also in B
So A βͺ B contains exactly the elements of B
(A's elements are already in B)
So A βͺ B contains exactly the elements of B
(A's elements are already in B)
A βͺ B = B
(When A is a subset of B, their union is just B)
(When A is a subset of B, their union is just B)
π‘ Key Rule: If A β B, then A βͺ B = B
4
Find the unions for A = {1,2,3,4}, B = {3,4,5,6}, C = {5,6,7,8}, D = {7,8,9,10}
(i) A βͺ B (ii) A βͺ C (iii) B βͺ C
(iv) B βͺ D (v) A βͺ B βͺ C (vi) A βͺ B βͺ D (vii) B βͺ C βͺ D
(iv) B βͺ D (v) A βͺ B βͺ C (vi) A βͺ B βͺ D (vii) B βͺ C βͺ D
π‘ Click to View Solution
1 Solutions
(i) A βͺ B = {1,2,3,4} βͺ {3,4,5,6}
All elements from both: 1,2,3,4,5,6
= {1, 2, 3, 4, 5, 6}
All elements from both: 1,2,3,4,5,6
= {1, 2, 3, 4, 5, 6}
(ii) A βͺ C = {1,2,3,4} βͺ {5,6,7,8}
No common elements
= {1, 2, 3, 4, 5, 6, 7, 8}
No common elements
= {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B βͺ C = {3,4,5,6} βͺ {5,6,7,8}
Common: 5, 6
= {3, 4, 5, 6, 7, 8}
Common: 5, 6
= {3, 4, 5, 6, 7, 8}
(iv) B βͺ D = {3,4,5,6} βͺ {7,8,9,10}
= {3, 4, 5, 6, 7, 8, 9, 10}
= {3, 4, 5, 6, 7, 8, 9, 10}
(v) A βͺ B βͺ C = ({1,2,3,4} βͺ {3,4,5,6}) βͺ {5,6,7,8}
= {1,2,3,4,5,6} βͺ {5,6,7,8}
= {1, 2, 3, 4, 5, 6, 7, 8}
= {1,2,3,4,5,6} βͺ {5,6,7,8}
= {1, 2, 3, 4, 5, 6, 7, 8}
(vi) A βͺ B βͺ D = {1,2,3,4,5,6} βͺ {7,8,9,10}
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B βͺ C βͺ D = {3,4,5,6,7,8} βͺ {7,8,9,10}
= {3, 4, 5, 6, 7, 8, 9, 10}
= {3, 4, 5, 6, 7, 8, 9, 10}
5
Find the intersection of each pair from Question 1
π‘ Click to View Solution
1 Understanding Intersection
Intersection (A β© B): Elements that are in BOTH A AND B
Only common elements
Only common elements
2 Solutions
(i) X β© Y = {1,3,5} β© {1,2,3}
Common elements: 1, 3
= {1, 3}
Common elements: 1, 3
= {1, 3}
(ii) A β© B = {a,e,i,o,u} β© {a,b,c}
Common: only 'a'
= {a}
Common: only 'a'
= {a}
(iii) A β© B = {3,6,9,12,...} β© {1,2,3,4,5}
Common: 3 (only 3 is in both)
= {3}
Common: 3 (only 3 is in both)
= {3}
(iv) A β© B = {2,3,4,5,6} β© {7,8,9}
No common elements
= Ο
No common elements
= Ο
(v) A β© Ο = {1,2,3} β© Ο
No elements in empty set
= Ο
No elements in empty set
= Ο
π‘ Key Property: A β© Ο = Ο (Intersection with empty set always gives empty set)
6
Find intersections: A = {3,5,7,9,11}, B = {7,9,11,13}, C = {11,13,15}, D = {15,17}
(i) A β© B (ii) B β© C (iii) A β© C β© D
(iv) A β© C (v) B β© D (vi) A β© (B βͺ C)
(vii) A β© D (viii) A β© (B βͺ D) (ix) (A β© B) β© (B βͺ C) (x) (A βͺ D) β© (B βͺ C)
(iv) A β© C (v) B β© D (vi) A β© (B βͺ C)
(vii) A β© D (viii) A β© (B βͺ D) (ix) (A β© B) β© (B βͺ C) (x) (A βͺ D) β© (B βͺ C)
π‘ Click to View Solution
1 Solutions
(i) A β© B = {3,5,7,9,11} β© {7,9,11,13}
Common: 7, 9, 11
= {7, 9, 11}
Common: 7, 9, 11
= {7, 9, 11}
(ii) B β© C = {7,9,11,13} β© {11,13,15}
Common: 11, 13
= {11, 13}
Common: 11, 13
= {11, 13}
(iii) A β© C β© D = (A β© C) β© D
A β© C = {3,5,7,9,11} β© {11,13,15} = {11}
{11} β© {15,17} = Ο
= Ο
A β© C = {3,5,7,9,11} β© {11,13,15} = {11}
{11} β© {15,17} = Ο
= Ο
(iv) A β© C = {3,5,7,9,11} β© {11,13,15}
Common: 11
= {11}
Common: 11
= {11}
(v) B β© D = {7,9,11,13} β© {15,17}
No common elements
= Ο
No common elements
= Ο
(vi) A β© (B βͺ C)
B βͺ C = {7,9,11,13} βͺ {11,13,15} = {7,9,11,13,15}
A β© {7,9,11,13,15} = {7,9,11}
= {7, 9, 11}
B βͺ C = {7,9,11,13} βͺ {11,13,15} = {7,9,11,13,15}
A β© {7,9,11,13,15} = {7,9,11}
= {7, 9, 11}
(vii) A β© D = {3,5,7,9,11} β© {15,17}
No common elements
= Ο
No common elements
= Ο
(viii) A β© (B βͺ D)
B βͺ D = {7,9,11,13} βͺ {15,17} = {7,9,11,13,15,17}
A β© {7,9,11,13,15,17} = {7,9,11}
= {7, 9, 11}
B βͺ D = {7,9,11,13} βͺ {15,17} = {7,9,11,13,15,17}
A β© {7,9,11,13,15,17} = {7,9,11}
= {7, 9, 11}
(ix) (A β© B) β© (B βͺ C)
A β© B = {7,9,11}, B βͺ C = {7,9,11,13,15}
{7,9,11} β© {7,9,11,13,15} = {7,9,11}
= {7, 9, 11}
A β© B = {7,9,11}, B βͺ C = {7,9,11,13,15}
{7,9,11} β© {7,9,11,13,15} = {7,9,11}
= {7, 9, 11}
(x) (A βͺ D) β© (B βͺ C)
A βͺ D = {3,5,7,9,11,15,17}
B βͺ C = {7,9,11,13,15}
Common: 7,9,11,15
= {7, 9, 11, 15}
A βͺ D = {3,5,7,9,11,15,17}
B βͺ C = {7,9,11,13,15}
Common: 7,9,11,15
= {7, 9, 11, 15}
7
Intersections with subsets: A = natural numbers, B = even naturals, C = odd naturals, D = primes
(i) A β© B (ii) A β© C (iii) A β© D (iv) B β© C
(v) B β© D (vi) C β© D
(v) B β© D (vi) C β© D
π‘ Click to View Solution
1 Solutions
(i) A β© B (natural β© even natural)
Even numbers are natural numbers
= B (all even natural numbers)
Even numbers are natural numbers
= B (all even natural numbers)
(ii) A β© C (natural β© odd natural)
Odd numbers are natural numbers
= C (all odd natural numbers)
Odd numbers are natural numbers
= C (all odd natural numbers)
(iii) A β© D (natural β© prime)
Prime numbers are natural numbers
= D (all prime numbers)
Prime numbers are natural numbers
= D (all prime numbers)
(iv) B β© C (even β© odd)
No number is both even AND odd
= Ο
No number is both even AND odd
= Ο
(v) B β© D (even β© prime)
Only 2 is both even and prime
= {2}
Only 2 is both even and prime
= {2}
(vi) C β© D (odd β© prime)
All odd primes: 3, 5, 7, 11, 13, ...
= {x : x is odd prime number}
All odd primes: 3, 5, 7, 11, 13, ...
= {x : x is odd prime number}
8
Which pairs are disjoint sets?
(i) {1, 2, 3, 4} and {x : x β β and 4 β€ x β€ 6}
(ii) {a, e, i, o, u} and {c, d, e, f}
(iii) {x : x is even integer} and {x : x is odd integer}
π‘ Click to View Solution
1 Understanding Disjoint Sets
Disjoint Sets: Sets that have NO common elements
A and B are disjoint if A β© B = Ο
A and B are disjoint if A β© B = Ο
2 Check each pair
(i) {1,2,3,4} and {4,5,6}
Common element: 4
{1,2,3,4} β© {4,5,6} = {4} β Ο
NOT disjoint β
Common element: 4
{1,2,3,4} β© {4,5,6} = {4} β Ο
NOT disjoint β
(ii) {a,e,i,o,u} and {c,d,e,f}
Common element: e
Intersection = {e} β Ο
NOT disjoint β
Common element: e
Intersection = {e} β Ο
NOT disjoint β
(iii) Even integers and Odd integers
No number is both even AND odd
Intersection = Ο
DISJOINT β
No number is both even AND odd
Intersection = Ο
DISJOINT β
π‘ Key: Sets are disjoint only if they have NO common elements (intersection is empty)
9
Find the difference: A = {3,6,9,12,15,18,21}, B = {4,8,12,16,20}, C = {2,4,6,8,10,12,14,16}, D = {5,10,15,20}
(i) A β B (ii) A β C (iii) A β D (iv) B β A
(v) C β A (vi) D β A (vii) B β C (viii) B β D
(ix) C β B (x) D β B (xi) C β D (xii) D β C
(v) C β A (vi) D β A (vii) B β C (viii) B β D
(ix) C β B (x) D β B (xi) C β D (xii) D β C
π‘ Click to View Solution
1 Understanding Difference
Difference (A β B): Elements in A but NOT in B
Remove all elements of B from A
Remove all elements of B from A
2 Solutions
(i) A β B = {3,6,9,12,15,18,21} β {4,8,12,16,20}
Remove 12 (only common element)
= {3, 6, 9, 15, 18, 21}
Remove 12 (only common element)
= {3, 6, 9, 15, 18, 21}
(ii) A β C = {3,6,9,12,15,18,21} β {2,4,6,8,10,12,14,16}
Remove 6, 12
= {3, 9, 15, 18, 21}
Remove 6, 12
= {3, 9, 15, 18, 21}
(iii) A β D = {3,6,9,12,15,18,21} β {5,10,15,20}
Remove 15
= {3, 6, 9, 12, 18, 21}
Remove 15
= {3, 6, 9, 12, 18, 21}
(iv) B β A = {4,8,12,16,20} β {3,6,9,12,15,18,21}
Remove 12
= {4, 8, 16, 20}
Remove 12
= {4, 8, 16, 20}
(v) C β A = {2,4,6,8,10,12,14,16} β {3,6,9,12,15,18,21}
Remove 6, 12
= {2, 4, 8, 10, 14, 16}
Remove 6, 12
= {2, 4, 8, 10, 14, 16}
(vi) D β A = {5,10,15,20} β {3,6,9,12,15,18,21}
Remove 15
= {5, 10, 20}
Remove 15
= {5, 10, 20}
(vii) B β C = {4,8,12,16,20} β {2,4,6,8,10,12,14,16}
Remove 4, 8, 12, 16
= {20}
Remove 4, 8, 12, 16
= {20}
(viii) B β D = {4,8,12,16,20} β {5,10,15,20}
Remove 20
= {4, 8, 12, 16}
Remove 20
= {4, 8, 12, 16}
(ix) C β B = {2,4,6,8,10,12,14,16} β {4,8,12,16,20}
Remove 4, 8, 12, 16
= {2, 6, 10, 14}
Remove 4, 8, 12, 16
= {2, 6, 10, 14}
(x) D β B = {5,10,15,20} β {4,8,12,16,20}
Remove 20
= {5, 10, 15}
Remove 20
= {5, 10, 15}
(xi) C β D = {2,4,6,8,10,12,14,16} β {5,10,15,20}
Remove 10
= {2, 4, 6, 8, 12, 14, 16}
Remove 10
= {2, 4, 6, 8, 12, 14, 16}
(xii) D β C = {5,10,15,20} β {2,4,6,8,10,12,14,16}
Remove 10
= {5, 15, 20}
Remove 10
= {5, 15, 20}
10
If X = {a, b, c, d} and Y = {f, b, d, g}, find X β Y, Y β X, and X β© Y
π‘ Click to View Solution
1 Solutions
(i) X β Y = {a,b,c,d} β {f,b,d,g}
Elements in X but not in Y: a, c
= {a, c}
Elements in X but not in Y: a, c
= {a, c}
(ii) Y β X = {f,b,d,g} β {a,b,c,d}
Elements in Y but not in X: f, g
= {f, g}
Elements in Y but not in X: f, g
= {f, g}
(iii) X β© Y = {a,b,c,d} β© {f,b,d,g}
Common elements: b, d
= {b, d}
Common elements: b, d
= {b, d}
11
If R is the set of real numbers and Q is the set of rational numbers, what is R β Q?
π‘ Click to View Solution
1 Logic
R β Q = real numbers that are NOT rational
These are called IRRATIONAL NUMBERS
These are called IRRATIONAL NUMBERS
R β Q = {x : x is an irrational number}
Examples: β2, Ο, e, β3, etc.
Examples: β2, Ο, e, β3, etc.
π‘ Key Fact: Every real number is either rational OR irrational (but not both)
12
State whether true or false and justify
(i) {2, 3, 4, 5} and {3, 6} are disjoint sets
(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets
π‘ Click to View Solution
1 Analyze each
(i) {2,3,4,5} β© {3,6} = {3} β Ο
Common element: 3
FALSE β
Common element: 3
FALSE β
(ii) {a,e,i,o,u} β© {a,b,c,d} = {a} β Ο
Common element: a
FALSE β
Common element: a
FALSE β
(iii) {2,6,10,14} β© {3,7,11,15} = Ο
No common elements
TRUE β
No common elements
TRUE β
(iv) {2,6,10} β© {3,7,11} = Ο
No common elements
TRUE β
No common elements
TRUE β
(i) FALSE (ii) FALSE (iii) TRUE (iv) TRUE
