Question

A subset of clients is described that the consultant could find using her database. HINT [See Example 4.] The clients who either do not owe her money, have done at least $10,000 worth of business with her, or have employed her in the last year. Write the subset in terms of A, B, and C. A ∪ B ∪ C A ∪ B' ∪ C' A' ∪ B ∪ C A' ∪ B' ∪ C' A' ∪ B ∪ C'

Answer 1

Write the subset in terms of A, B, and C.

1. A ∪ B ∪ C

2. A ∪ B' ∪ C'

3. A' ∪ B ∪ C

4. A' ∪ B' ∪ C'

5. A' ∪ B ∪ C'

Answer:

1. A ∪ B ∪ C = {Acme, Brothers, Crafts, Dion, Effigy, Global, Hilbert}

2. A ∪ B' ∪ C' = {Acme, Brothers, Craft, Dion, Effigy, Floyd, Global, Hilbert}

3. A' ∪ B ∪ C = {Acme, Brothers, Crafts, Dion, Effigy, Floyd, Global}

4. A' ∪ B' ∪ C' = {Brothers, Dion, Effigy, Floyd, Gilbert, Hilbert}

5. A' ∪ B ∪ C' = {Acme, Brothers, Crafts, Dion, Floyd, Hilbert}

Step-by-step explanation:

Given

From the hint in the question above

A = Set of all clients who do not owe her money.

A = {Acme, Craft, Dion, Effigy, Global, Hilbert}

B = Set of all clients that have done at least $10,000 worth of business with her

B = {Acme, Brothers, Crafts, Dion}

C = Set of all clients who have employed her in the last year

C = {Acme, Crafts, Effigy, Global}

Let U = The Universal Set = {Acme, Brothers, Crafts, Dion, Effigy, Floyd, Global, Hilbert}

1. A ∪ B ∪ C :

This means the set of every elements present in A, B and C without repetition of any element of the set

Given that

A = {Acme, Craft, Dion, Effigy, Global, Hilbert}

B = {Acme, Brothers, Crafts, Dion}

C = {Acme, Crafts, Effigy, Global}

A ∪ B ∪ C = {Acme, Brothers, Crafts, Dion, Effigy, Global, Hilbert}

2. A ∪ B' ∪ C'

Given that

A = {Acme, Craft, Dion, Effigy, Global, Hilbert}

B' = Sets of all elements in the universal set but not in B

So, B' = {Effigy, Floyd, Global, Hilbert}

C' = Sets of all elements in the universal set but not in C

So, C' = {Brothers, Dion, Floyd, Hilbert}

A ∪ B' ∪ C' = {Acme, Brothers, Craft, Dion, Effigy, Floyd, Global, Hilbert}

3. A' ∪ B ∪ C

A' = Sets of all elements in the universal set but not in A

So, A' = {Brothers, Floyd}

B = {Acme, Brothers, Crafts, Dion}

C = {Acme, Crafts, Effigy, Global}

A' ∪ B ∪ C = {Acme, Brothers, Crafts, Dion, Effigy, Floyd, Global}

4. A' ∪ B' ∪ C'

A' = Sets of all elements in the universal set but not in A

So, A' = {Brothers, Floyd}

B' = Sets of all elements in the universal set but not in B

So, B' = {Effigy, Floyd, Global, Hilbert}

C' = Sets of all elements in the universal set but not in C

So, C' = {Brothers, Dion, Floyd, Hilbert}

A' ∪ B' ∪ C' = {Brothers, Dion, Effigy, Floyd, Gilbert, Hilbert}

5. A' ∪ B ∪ C'

A' = Sets of all elements in the universal set but not in A

So, A' = {Brothers, Floyd}

B = {Acme, Brothers, Crafts, Dion}

C' = Sets of all elements in the universal set but not in C

So, C' = {Brothers, Dion, Floyd, Hilbert}

A' ∪ B ∪ C' = {Acme, Brothers, Crafts, Dion, Floyd, Hilbert}