Explanation:
Let A = {1, 2, 3}, B = {3, 4, 5}, and U = {1, 2, 3, 4, 5}.
a) We have:
(A∪B)′ = {1, 2, 4, 5} (taking the complement of the union of A and B)
A′∩B′ = {4, 5} (taking the intersection of the complements of A and B)
We can verify that (A∪B)′ = A′∩B′ by showing that both sets contain exactly the same elements, which they do in this case.
Another example: Let A = {a, b, c}, B = {c, d, e}, and U = {a, b, c, d, e}.
Then:
(A∪B)′ = {d, e}
A′∩B′ = {d, e}
Again, we can see that both sets contain exactly the same elements, so (A∪B)′ = A′∩B′ is true.
To prove this using Venn diagrams, we can draw a Venn diagram for the sets A, B, and (A∪B)′, and shade the region that represents the complement of the union of A and B. We can also draw a Venn diagram for A′, B′, and their intersection, and shade the region that represents this intersection. Then we can visually compare the two shaded regions to see that they are the same.
b) We have:
(A∩B)′ = {1, 2, 4, 5} (taking the complement of the intersection of A and B)
A′∪B′ = {1, 2, 4, 5} (taking the union of the complements of A and B)
We can see that (A∩B)′ = A′∪B′ in this case.
Another example: Let A = {a, b, c}, B = {c, d, e}, and U = {a, b, c, d, e}.
Then:
(A∩B)′ = {a, b, d, e}
A′∪B′ = {a, b, d, e}
Again, we can see that both sets contain exactly the same elements, so (A∩B)′ = A′∪B′ is true.
To prove this using Venn diagrams, we can draw a Venn diagram for the sets A, B, and their intersection, and shade the region that represents this intersection. We can also draw a Venn diagram for A′, B′, and their union, and shade the region that represents this union. Then we can visually compare the two shaded regions to see that they are the same.