Mathematics
Let U = {a, b, c, d, e, f, g} be the universal set and let its subsets be A = {a, b, d, e} and B = {b, e, g}.
Verify that:
(i) (A ∪ B)' = (A' ∩ B')
(ii) (A ∩ B)' = (A' ∪ B')
Answer
Given:
Universal set U = {a, b, c, d, e, f, g}
Subset A = {a, b, d, e}
Subset B = {b, e, g}
(i) (A ∪ B)' = (A' ∩ B')
First let us find A ∪ B:
A ∪ B = {a, b, d, e} ∪ {b, e, g} = {a, b, d, e, g}
(A ∪ B)' = Elements in U which are not in (A ∪ B).
LHS = (A ∪ B)' = U - (A ∪ B)
LHS = {a, b, c, d, e, f, g} - {a, b, d, e, g}
LHS = {c, f}
Now, find A' and B':
A' = U - A
A' = {a, b, c, d, e, f, g} - {a, b, d, e} = {c, f, g}
B' = U - B
B' = {a, b, c, d, e, f, g} - {b, e, g} = {a, c, d, f}
RHS = (A' ∩ B') = {c, f, g} ∩ {a, c, d, f}
RHS = {c, f}
Since LHS = RHS,
∴ The statement (A ∪ B)' = (A' ∩ B') is verified.
(ii) (A ∩ B)' = (A' ∪ B')
First let us find A ∩ B:
A ∩ B = {a, b, d, e} ∩ {b, e, g} = {b, e}
LHS = (A ∩ B)' = U - (A ∩ B).
LHS = (A ∩ B)' = {a, b, c, d, e, f, g} - {b, e}
LHS = {a, c, d, f, g}
Now, find A' and B':
A' = U - A
A' = {a, b, c, d, e, f, g} - {a, b, d, e} = {c, f, g}
B' = U - B
B' = {a, b, c, d, e, f, g} - {b, e, g} = {a, c, d, f}
RHS = (A' ∪ B') = {c, f, g} ∪ {a, c, d, f}
RHS = {a, c, d, f, g}
Since LHS = RHS,
∴ The statement (A ∩ B)' = (A' ∪ B') is verified.
Related Questions
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')
Let U = {3, 6, 9, 12, 15, 18, 21, 24} be the universal set and let A = {6, 12, 18, 24} be its subset.
Verify that:
(i) A ∪ A = A
(ii) A ∩ A = A
(iii) A ∩ A' = Φ
(iv) A ∪ A' = U
(v) (A')' = A
Let A and B be two sets such that n(A) = 52, n(B) = 60 and n(A ∩ B) = 16. Draw a Venn diagram and find :
(i) n(A ∪ B)
(ii) n(A - B)
(iii) n(B - A)