Question

Let set U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, set A = {2, 4, 6, 8}, and set B = {1, 3, 5, 7, 9}. Prove that (A $\cap$ B)’ = A’ $\cup$ B.’

Answer 1

(A $\cap$ B)’ = A’ $\cup$ B.’

We will evaluate both the left-hand side and the right-hand side separately.

Left-Hand Side:

(A $\cap$ B)’

Let’s first the intersection of A and B given as:

(A $\cap$ B)

So,

(A $\cap$ B) = {2, 4, 6, 8} $\cap$ {1, 3, 5, 7, 9}

(A $\cap$ B) = { }

Now, let’s find the complement of the intersection, which is given as:

(A $\cap$ B)’ = U − (A $\cap$ B)

(A $\cap$ B)’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} − { }

(A $\cap$ B)’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Now, let’s evaluate the right-hand side.

Right-Hand Side:

A’ $\cap$ B’

Let’s first find the complements:

The complement of A is:

A’ = U − A

A’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} − {2, 4, 6, 8}

A’ = {0, 1, 3, 5, 7, 9, 10}

The complement of B is:

B’ = U − B

B’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} − {1, 3, 5, 7, 9}

B’ = {0, 2, 4, 6, 8, 10}

Now, the union of the complements is given as:

A’ $\cup$ B’ = {0, 1, 3, 5, 7, 9, 10} $\cup$ {0, 2, 4, 6, 8, 10}

A’ $\cup$ B’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

This proves that left-hand side = right-hand side and hence:

(A $\cap$ B)’ = A’ $\cap$ B.’