Mathematics
If A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6} ; verify :
(i) A - (B ∪ C) = (A - B) ∩ (A - C)
(ii) A - (B ∩ C) = (A - B) ∪ (A - C).
Sets
2 Likes
Answer
A = {1, 2, 3, 4, 5}
B = {2, 4, 6, 8}
C = {3, 4, 5, 6}
(i) A - (B ∪ C) = (A - B) ∩ (A - C)
Taking LHS : A - (B ∪ C)
B ∪ C - contains all the elements in set B and C.
B ∪ C = {2, 4, 6, 8} ∪ {3, 4, 5, 6}
B ∪ C = {2, 3, 4, 5, 6, 8}
A - (B ∪ C) - contains all the elements which are in set A but not in B ∪ C.
A - (B ∪ C) = {1, 2, 3, 4, 5} - {2, 3, 4, 5, 6, 8}
A - (B ∪ C) = {1} ……………(1)
Taking RHS : (A - B) ∩ (A - C)
A - B - contains all the elements which are in set A but not in B.
A - B = {1, 2, 3, 4, 5} - {2, 4, 6, 8}
A - B = {1, 3, 5}
A - C - contains all the elements which are in set A but not in C.
A - C = {1, 2, 3, 4, 5} - {3, 4, 5, 6}
A - C = {1, 2}
(A - B) ∩ (A - C) - contains all the common elements in set (A - B) and (A - C).
(A - B) ∩ (A - C) = {1, 3, 5} ∩ {1, 2}
(A - B) ∩ (A - C) = {1} ……………(2)
From (1) and (2), we can see
LHS = RHS
Hence, A - (B ∪ C) = (A - B) ∩ (A - C)
(ii) A - (B ∩ C) = (A - B) ∪ (A - C).
Taking LHS : A - (B ∩ C)
B ∩ C - contains all the common elements in set B and C.
B ∩ C = {2, 4, 6, 8} ∩ {3, 4, 5, 6}
B ∩ C = {4, 6}
A - (B ∩ C) - contains all the elements which are in set A but not in B ∩ C.
A - (B ∩ C) = {1, 2, 3, 4, 5} - {4, 6}
A - (B ∩ C) = {1, 2, 3, 5} ……………(3)
Taking RHS : (A - B) ∪ (A - C)
A - B - contains all the elements which are in set A but not in B.
A - B = {1, 2, 3, 4, 5} - {2, 4, 6, 8}
A - B = {1, 3, 5}
A - C - contains all the elements which are in set A but not in C.
A - C = {1, 2, 3, 4, 5} - {3, 4, 5, 6}
A - C = {1, 2}
(A - B) ∪ (A - C) - contains all the elements in set (A - B) and (A - C).
(A - B) ∪ (A - C) = {1, 3, 5} ∪ {1, 2}
(A - B) ∪ (A - C) = {1, 2, 3, 5} ……………(4)
From (3) and (4), we can see
LHS = RHS
Hence, A - (B ∩ C) = (A - B) ∪ (A - C)
Answered By
2 Likes
Related Questions
If P = {factors of 36} and Q = {factors of 48} ; find :
(i) P ∪ Q
(ii) P ∩ Q
(iii) Q - P
(iv) P' ∩ Q
If A = {6, 7, 8, 9}, B = {4, 6, 8, 10} and C ={x : x ∈ N : 2 < x ≤ 7}; find :
(i) A - B
(ii) B - C
(iii) B - (A - C)
(iv) A - (B ∪ C)
(v) B - (A ∩ C)
(vi) B - B
Given A = {x ∈ N : x < 6}, B = {3, 6, 9} and C = {x ∈ N : 2x - 5 ≤ 8}. Show that :
(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
If n(A) = 30, n(B) = 20 and n(A ∪ B) = 36, find n(A ∩ B).