LetA, B, C and D be the following sets:

A={red, blue, green, purple}

B={red, red, green}

C={red,{green}, red,{red, green},{green, green, red}}

D={blue, blue, blue, green, green, purple}

Let the universal set for A, B, C, D be defined by:

U={red, blue, purple, green}⋃P ({red, blue, purple, green}) Provide the answer to the following quesitons about the above sets:

(A) : Is B ∈ A?

(B) : Is B ∈ C?

(C) : Is B ∈ P (C) ?

(D) What is |A|?

(E) What is |C|?

(F) What is |D|?

(a) No

(b) Yes

(c) No

(d) 4

(e) 3

(f) 3

Step-by-step explanation:

A = {red, blue, green, purple}

B = {red, red, green}

C = {red,{green}, red,{red, green},{green, green, red}}

D = {blue, blue, blue, green, green, purple}

U = {red, blue, purple, green}∪ P ({red, blue, purple, green})

By removing duplicate elements, the sets become

A = {red, blue, green, purple}

B = {red, green}

C = {red, {green}, {red, green}}

D = {blue, green, purple}

U = {red, blue, purple, green} ∪ P ({red, blue, purple, green})

(a) B ∉ A because B is a set while A contains elements that are not sets, even thought the elements of B are also in A. In fact, B is a subset of A.

(b) B ∈ C because {red, green} is a member of C.

(c) P (C) is the power set of C, the set of all subsets of C. The power set of C is

P (C) = {{}, {red}, {{green}}, {{red, green}}, {red, {green}}, {red, {red, green}}, {{green}, {red, green}}, {red, {green}, {red, green}}}

It is seen that B ∉ P (C).

(d) |A| is the cardinality of A or the number of distinct members of A.

Therefore, |A| = 4

(e) |C| = 3 (Note that the elements in the member set are not counted)

(f) |D| = 3