Question

Let U = {a, b, c, d, e, f, g, h, i, j, k}.

Let A = {a, c, d, f, g, i}
Let B = {b, c, d, f, g}
Let C={a, b, f, i, j}.
Determine (A' UC) U (A n B).

Answer 1

see below↓

Explanation:

A' represents the complement of set A, which is {b, e, h, j, k}.

UC represents the union of sets U and C, which is {a, b, c, f, i, j}.

A n B represents the intersection of sets A and B, which is {c, d, f, g}.

Now, let's put it all together:

(A' UC) U (A n B) = ({b, e, h, j, k} {a, b, c, f, i, j}) U {c, d, f, g}

= {a, b, c, e, f, g, h, i, j, k} U {c, d, f, g}

= {a, b, c, d, e, f, g, h, i, j, k}

So, the result of the expression is {a, b, c, d, e, f, g, h, i, j, k}.

`\:`

Answer 2

(A' ∪ C) ∪ (A ∩ B) = {a, b, c, d, e, f, g, h, i, j, k}.

Explanation:

Note:

• A' is the complement of set A, which contains all elements in the universal set U that are not in A.

• A ∪ C is a union of sets A and C, which contains all elements that are in either A or C.

• A ∩ B is the intersection of sets A and B, which contains all elements that are in both A and B.

`\hrulefill`

For the Question:

Let's find the values for each set:

• A' (complement of A): {b, e, h, j, k}
• A ∪ C (union of A and C): {a, b, c, d, f, g, i, j}
• A ∩ B (intersection of A and B): {c, d, f, g}

Now, let's proceed with the given expression:

(A' ∪ C) ∪ (A ∩ B)

Evaluate (A' ∪ C):

(A' ∪ C) = {b, e, h, j, k} ∪ {a, b, f, i, j}

(A' ∪ C) = {a, b, e, f, h, i, j, k}

Now, union this result with (A ∩ B):

{a, b, e, f, h, i, j, k} ∪ {c, d, f, g}

The resulting set will be:

{a, b, c, d, e, f, g, h, i, j, k}

So, (A' ∪ C) ∪ (A ∩ B) = {a, b, c, d, e, f, g, h, i, j, k}, which is the universal set U.