Question

A freelance computer consultant keeps a database of her clients, which contains the names S = {Acme, Bakers, Cores, Dual, Energy, Flavour, Global, Hilbert}. These clients owe her money:
A = {Acme, Cores, Energy, Global}.

These clients have done at least $2,000 worth of business with her:

B = {Acme, Bakers, Cores, Dual}.

These clients have employed her in the last year:

C = {Acme, Cores, Dual, Energy, Global, Hilbert}.

A subset of clients is described that the consultant could find using her database. HINT [See Example 4.]
The clients who do not owe her money or have employed her in the last year. Write the subset in terms of A, B, and C.

A. A' ∩ B
B. A' ∩ C
C. A ∪ C
D. A ∩ B
E. A' ∪ C

List the clients in that subset.

A. {Flavour, Bakers}
B. {Acme, Energy, Cores, Global, Bakers, Dual, Flavour, Hilbert}
C. {Acme, Energy, Cores, Global}
D. ∅
E. {Acme, Cores}

Answer 1

We want clients that do not owe her money or have employed her in the last year. In set terminology, the or operator is represented by the union of two sets.

So, we're looking for the union between the subsets of clients that don't owe money and clients that have employes in the last year.

The first subset is A', because we're looking for the negation of its condition.

The second subset, by definition, is exactly C.

So, the answer is A' ∪ C.

To list the customer, we have:

A' = {Bakers, Dual, Flavour, Hilbert}

C = {Acme, Cores, Dual, Energy, Global, Hilbert}

So, their union is composed by all elements belonging to A' or E (or both), without repetitions:

A' ∪ C = {Acme, Bakers, Cores, Dual, Energy, Flavour, Global, Hilbert} = S