Mathematics
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
Sets
1 Like
Answer
Given:
Universal set U = {3, 6, 9, 12, 15, 18, 21, 24}
Subset A = {6, 12, 18, 24}
(i) A ∪ A = A
LHS = A ∪ A = {6, 12, 18, 24} ∪ {6, 12, 18, 24}
LHS = {6, 12, 18, 24} [since repeated elements are written once]
RHS = A = {6, 12, 18, 24}
Since LHS = RHS,
∴ The statement A ∪ A = A is verified.
(ii) A ∩ A = A
LHS = A ∩ A = {6, 12, 18, 24} ∩ {6, 12, 18, 24}
LHS = {6, 12, 18, 24}
RHS = A = {6, 12, 18, 24}
Since LHS = RHS,
∴ The statement A ∩ A = A is verified.
(iii) A ∩ A' = Φ
A' = U - A = {3, 6, 9, 12, 15, 18, 21, 24} - {6, 12, 18, 24} = {3, 9, 15, 21}
LHS = A ∩ A' = {6, 12, 18, 24} ∩ {3, 9, 15, 21}
LHS = Φ
RHS = Φ
Since LHS = RHS,
∴ The statement A ∩ A' = Φ is verified.
(iv) A ∪ A' = U
A' = {3, 9, 15, 21} [From previous step]
LHS = A ∪ A' = {6, 12, 18, 24} ∪ {3, 9, 15, 21}
LHS = {3, 6, 9, 12, 15, 18, 21, 24}
RHS = U = {3, 6, 9, 12, 15, 18, 21, 24}
Since LHS = RHS,
∴ The statement A ∪ A' = U is verified.
(v) (A')' = A
We know A' = {3, 9, 15, 21}
LHS = (A')' = U - A' = {3, 6, 9, 12, 15, 18, 21, 24} - {3, 9, 15, 21}
LHS = {6, 12, 18, 24}
RHS = A = {6, 12, 18, 24}
Since LHS = RHS,
∴ The statement (A')' = A is verified.
Answered By
1 Like
Related Questions
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 = {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')
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)
Let P and Q be two sets such that n(P ∪ Q) = 70, n(P) = 45 and n(Q) = 38. Draw a Venn diagram and find :
(i) n(P ∩ Q)
(ii) n(P - Q)
(iii) n(Q - P)