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

use this formula:

p(a\b) = p(a and b)/p(b).

take it from here.

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12