Question

prove that a union b whole dash equal to a dash intersection b dash

Answer 1

Hi
Check this out.
Second image is of last step.

Answer 2

Answer:

We have to prove that  (A ∪ B)' = A' ∩ B'

Let x be an arbitrary element of (A ∪ B)'

⇒ x ∈ (A ∪ B)'

⇒ x ∉ A ∪ B

⇒ x ∉ A and x ∉ B

⇒ x ∈ A' and x ∈ B'

⇒ x ∈ (A' ∩ B')

Therefore, (A ∪ B)' ⊆ (A' ∩ B')                      ....................(1)

Now, to prove  (A' ∩ B') ⊆ (A ∪ B)'

Let y be an arbitrary element of (A' ∩ B')

⇒ x ∈ (A' ∩ B')

⇒ x ∈ A' and x ∈ B'

⇒ x ∉ A and x ∉ B

⇒ x ∉ (A ∪ B)

⇒ x∈ (A ∪ B)'

Therefore, (A' ∩ B') ⊆ (A ∪ B)'                     ....................(2)

From equation (1) and (2), (A ∪ B)' = A' ∩ B'

Hence proved