Mathematics
Let A = {x : x is a multiple of 2, x < 15}, B = {x : x is a multiple of 3, x < 20}, C = {x : x is a prime, x < 20}.
(i) Write each one of the sets A, B, C in Roster form.
(ii) Find:
(a) A ∪ B
(b) A ∪ C
(c) B ∪ C
(d) A ∩ B
(e) A ∩ C
(f) B ∩ C
Answer
(i) Write each one of the sets A, B, C in Roster form.
Given:
A = {x : x is a multiple of 2, x < 15}
Multiples of 2 less than 15 are: 2, 4, 6, 8, 10, 12, 14.
A = {2, 4, 6, 8, 10, 12, 14}
B = {x : x is a multiple of 3, x < 20}
Multiples of 3 less than 20 are: 3, 6, 9, 12, 15, 18.
B = {3, 6, 9, 12, 15, 18}
C = {x : x is a prime, x < 20}
Prime numbers between 1 and 20 are: 2, 3, 5, 7, 11, 13, 17, 19.
C = {2, 3, 5, 7, 11, 13, 17, 19}
∴ A = {2, 4, 6, 8, 10, 12, 14}, B = {3, 6, 9, 12, 15, 18} and C = {2, 3, 5, 7, 11, 13, 17, 19}
(ii) Find:
(a) A ∪ B
Given:
A = {2, 4, 6, 8, 10, 12, 14}
B = {3, 6, 9, 12, 15, 18}
A ∪ B = {2, 4, 6, 8, 10, 12, 14} ∪ {3, 6, 9, 12, 15, 18} = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 18}
∴ A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 18}
(b) A ∪ C
Given:
A = {2, 4, 6, 8, 10, 12, 14}
C = {2, 3, 5, 7, 11, 13, 17, 19}
A ∪ C = {2, 4, 6, 8, 10, 12, 14} ∪ {2, 3, 5, 7, 11, 13, 17, 19} = {2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 19}
∴ A ∪ C = {2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 19}
(c) B ∪ C
Given:
B = {3, 6, 9, 12, 15, 18}
C = {2, 3, 5, 7, 11, 13, 17, 19}
B ∪ C = {3, 6, 9, 12, 15, 18} ∪ {2, 3, 5, 7, 11, 13, 17, 19} = {2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19}
∴ B ∪ C = {2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19}
(d) A ∩ B
Given:
A = {2, 4, 6, 8, 10, 12, 14}
B = {3, 6, 9, 12, 15, 18}
A ∩ B = {2, 4, 6, 8, 10, 12, 14} ∩ {3, 6, 9, 12, 15, 18} = {6, 12}
∴ A ∩ B = {6, 12}
(e) A ∩ C
Given:
A = {2, 4, 6, 8, 10, 12, 14}
C = {2, 3, 5, 7, 11, 13, 17, 19}
A ∩ C = {2, 4, 6, 8, 10, 12, 14} ∩ {2, 3, 5, 7, 11, 13, 17, 19} = {2}
∴ A ∩ C = {2}
(f) B ∩ C
Given:
B = {3, 6, 9, 12, 15, 18}
C = {2, 3, 5, 7, 11, 13, 17, 19}
B ∩ C = {3, 6, 9, 12, 15, 18} ∩ {2, 3, 5, 7, 11, 13, 17, 19} = {3}
∴ B ∩ C = {3}
Related Questions
Let P = {x : x is a factor of 18} and Q = {x : x is a factor of 24}.
(i) Write each one of P and Q in Roster form.
(ii) Find:
(a) P ∪ Q
(b) P ∩ Q
Let A = {a, b, c}, B = {b, d, e} and C = {e, f, g}, verify that :
(i) A ∪ B = B ∪ A
(ii) (A ∪ B) ∪ C = A ∪ (B ∪ C)
(iii) A ∩ B = B ∩ A
(iv) (A ∩ B) ∩ C = A ∩ (B ∩ C)
Let A = {b, d, e, f}, B = {c, d, g, h} and C = {e, f, g, h}. Find :
(i) A - B
(ii) B - C
(iii) C - A
(iv) (A - B) ∪ (B - A)
(v) (B - C) ∪ (C - B)
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set and let A = {2, 3, 4, 5, 6} and B = {3, 5, 7, 8} be its subsets.
Find:
(i) A'
(ii) B'
(iii) A' ∩ B'
(iv) A' ∪ B'
Verify that:
(v) (A ∪ B)' = (A' ∩ B')
(vi) (A ∩ B)' = (A' ∪ B')