Question

LetAbe the set$\mathbf{\{}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{\}}$and$\mathbf{B}$be the set$\mathbf{\{}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{\}}$.

1. IsAa subset ofB?
2. IsBa subset ofA?
3. What is$\mathbf{A}\mathbf{\cup }\mathbf{B}$?
4. What is$\mathbf{A}\mathbf{\cap }\mathbf{B}$?
5. What is$\mathbf{A}\mathbf{\times }\mathbf{B}$?
6. What is the power set ofB ?

Short answer

a. No, A is not a subset of B.

b. Yes, B is a subset of A.

c. $\mathrm{A}\cup \mathrm{B}=\{\mathrm{x},\mathrm{y},\mathrm{z}\}$

d. $\mathrm{A}\cap \mathrm{B}=\mathrm{B}$

e. $\mathrm{A}\times \mathrm{B}=\{(\mathrm{x},\mathrm{y}),(\mathrm{x},\mathrm{y}),(\mathrm{y},\mathrm{x}),(\mathrm{y},\mathrm{y}),(\mathrm{z},\mathrm{x}),(\mathrm{z},\mathrm{y})\}$

f. Power set of B is$\{\mathrm{\varphi },\left\{\mathrm{x}\right\},\left\{\mathrm{y}\right\},\{\mathrm{x},\mathrm{y}\}\}$

Step-by-step solution

Explain the concepts in a set

A set is amathematical model for the collection of different things and containselements or members, which are numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other set.

The union of two setsXandYis adequate to the set of elements that are present in setX, setY, or in both the setsXandY. This operation is often represented as;$\mathbf{X}\mathbf{\cup }\mathbf{Y}\mathbf{=}\mathbf{\{}\mathbf{a}\mathbf{:}\mathbf{a}\mathbf{\in }\mathbf{X}\mathbf{ora}\mathbf{\in }\mathbf{Y}\mathbf{\}}$_.

The set of elements shared by each of the given sets is found at the intersection of two or more given sets. The symbol $\mathbf{\cap }$ denotes the point where two sets intersect.

Explain the given statement being incorrect

a.

According to the given information,$\mathrm{A}=\{\mathrm{x},\mathrm{y},\mathrm{z}\},\mathrm{B}=\{\mathrm{x},\mathrm{y}\}$ .Z is an additional element in A that is absent from B.

Therefore, A is not termed as a subset of B.

Explain the given statement being correct

b.

According to the given information,$\mathrm{A}=\{\mathrm{x},\mathrm{y},\mathrm{z}\},\mathrm{B}=\{\mathrm{x},\mathrm{y}\}$.Here, every member ofBis a member ofAalso. Then,Bis said to be a proper subset ofA.
Hence, B is termed as a subset of A.

Find A∪B

c.

Let us consider the given sets,$\mathrm{A}=\{\mathrm{x},\mathrm{y},\mathrm{z}\},\mathrm{B}=\{\mathrm{x},\mathrm{y}\}$ .

$\begin{array}{c}\mathrm{A}\cup \mathrm{B}=\{\mathrm{x},\mathrm{y},\mathrm{z}\}\cup \{\mathrm{x},\mathrm{y}\}\\ =\{\mathrm{x},\mathrm{y},\mathrm{z}\}\\ =\mathrm{A}\end{array}$

So, $\mathrm{A}\cup \mathrm{B}$ is A.

Find A∩B

d.

Let us consider the given sets,$\mathrm{A}=\{\mathrm{x},\mathrm{y},\mathrm{z}\},\mathrm{B}=\{\mathrm{x},\mathrm{y}\}$

$\begin{array}{c}\mathrm{A}\cap \mathrm{B}=\{\mathrm{x},\mathrm{y},\mathrm{z}\}\cap \{\mathrm{x},\mathrm{y}\}\\ =\{\mathrm{x},\mathrm{y}\}\\ =\mathrm{B}\end{array}$

Accordingly, $\mathrm{A}\cap \mathrm{B}$is B.

Find the cartesian product.

e.

The Cartesian product of two sets A and B, denoted by A × B, is the set of all (x, y) such that a is a member of set A and b is a member of set B.

Hence,$\mathrm{A}\times \mathrm{B}=\{(\mathrm{x},\mathrm{y}),(\mathrm{x},\mathrm{y}),(\mathrm{y},\mathrm{x}),(\mathrm{y},\mathrm{y}),(\mathrm{z},\mathrm{x}),(\mathrm{z},\mathrm{y})\}$

Find the power set of B.

f.

Power Set is the combination of all subsets (including empty set) of a set.

Thus, the Power set of B is $\{\mathrm{\varphi },\left\{\mathrm{x}\right\},\left\{\mathrm{y}\right\},\{\mathrm{x},\mathrm{y}\}\}$.