Chapter 6.1, Problem 1E

a.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{\mathbb{Z}},-)$

Calculation:

Consider the set of integers

  $\mathrm{\mathbb{Z}}=\{\mathrm{...}-3,-2,-1,0,1,2,\mathrm{3....}\}$

Consider two elements from the given set.

The subtraction of two integers again gives an integer.

Therefore, $(\mathrm{\mathbb{Z}},-)$ is closed under subtraction.

Hence $(\mathrm{\mathbb{Z}},-)$ is the algebraic structure.

Associative part and commutative part:

It is known that the operation $-$ is neither commutative nor associative, hence $(\mathrm{\mathbb{Z}},-)$ is neither commutative nor associative.

b.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{\mathbb{Z}},÷)$

Calculation:

Consider the set of integers

  $\mathrm{\mathbb{Z}}=\{\mathrm{...}-3,-2,-1,0,1,2,\mathrm{3....}\}$

Consider two elements from the given set.

The division of two integers like $1÷2=0.5$ does not give an integer.

Therefore, $(\mathrm{\mathbb{Z}},÷)$ is not closed under division.

Hence $(\mathrm{\mathbb{Z}},÷)$ is not the algebraic structure.

Associative part and commutative part:

It is known that the operation $÷$ is neither commutative nor associative, hence $(\mathrm{\mathbb{Z}},÷)$ is neither commutative nor associative.

c.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{ℝ},-)$

Calculation:

Consider the set of real numbers $\mathrm{ℝ}$ .

Consider two elements from the given set.

The subtraction of two real numbers again gives a real number.

Therefore, $(\mathrm{ℝ},-)$ is closed under subtraction.

Hence $(\mathrm{ℝ},-)$ is the algebraic structure.

Associative part and commutative part:

It is known that the operation $-$ is neither commutative nor associative, hence $(\mathrm{ℝ},-)$ is neither commutative nor associative.

d.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{ℝ},÷)$

Calculation:

Consider the set of real numbers $\mathrm{ℝ}$ .

Consider two elements from the given set.

The division of two real numbers again gives a real number.

Therefore, $(\mathrm{ℝ},÷)$ is closed under division.

Hence $(\mathrm{ℝ},÷)$ is the algebraic structure.

Associative part and commutative part:

It is known that the operation $÷$ is neither commutative nor associative, hence $(\mathrm{ℝ},÷)$ is neither commutative nor associative.

e.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{\mathbb{N}},-)$

Calculation:

Consider the set of natural numbers

  $\mathrm{\mathbb{N}}=\{1,2,\mathrm{3....}\}$

Consider two elements example $1,2$ from the given set.

The subtraction of these two $1-2=-1\notin \mathrm{\mathbb{N}}$ .

Therefore, $(\mathrm{\mathbb{N}},-)$ is not closed under subtraction.

Hence $(\mathrm{\mathbb{N}},-)$ is not the algebraic structure.

Associative part and commutative part:

It is known that the operation $-$ is neither commutative nor associative, hence $(\mathrm{\mathbb{N}},-)$ is neither commutative nor associative.

f.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{\mathbb{Q}},÷)$

Calculation:

Consider the set of rational numbers $\mathrm{\mathbb{Q}}$ .

Since $\mathrm{a}÷0$ is not defined.

Therefore, $(\mathrm{\mathbb{Q}},÷)$ is not closed under division.

Hence $(\mathrm{\mathbb{Q}},÷)$ is not the algebraic structure.

Associative part and commutative part:

It is known that the operation $÷$ is neither commutative nor associative, hence $(\mathrm{\mathbb{Q}},÷)$ is neither commutative nor associative.

g.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{\mathbb{Q}}-\{0\},÷)$

Calculation:

Consider the set of rational numbers $\mathrm{\mathbb{Q}}$ .

Since $0$ is not included in the set.

Therefore, $(\mathrm{\mathbb{Q}},÷)$ is closed under division.

Hence $(\mathrm{\mathbb{Q}},÷)$ is the algebraic structure.

Associative part and commutative part:

It is known that the operation $÷$ is neither commutative nor associative, hence $(\mathrm{\mathbb{Q}},÷)$ is neither commutative nor associative.

h.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{ℝ}-\mathrm{\mathbb{Q}},\cdot )$

Calculation:

As $\mathrm{ℝ}=\mathrm{\mathbb{Q}}\cup {\mathrm{\mathbb{Q}}}^{\mathrm{c}}$ thus $\mathrm{ℝ}-\mathrm{\mathbb{Q}}={\mathrm{\mathbb{Q}}}^{\mathrm{c}}$

Therefore, $(\mathrm{ℝ}-\mathrm{\mathbb{Q}},\cdot )$ is not closed under multiplication.

Hence $(\mathrm{ℝ}-\mathrm{\mathbb{Q}},\cdot )$ is not the algebraic structure.

i.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{P}(\mathrm{A}),\cap )$

Calculation:

Consider the power set of any set $\mathrm{P}(\mathrm{A})$

Consider two elements from the given set.

The intersection of two elements will give the elements that belongs to power set.

Therefore, $(\mathrm{P}(\mathrm{A}),\cap )$ is closed under intersection.

Hence $(\mathrm{P}(\mathrm{A}),\cap )$ is the algebraic structure.

Associative part and commutative part:

It is known that the operation $\cap$ is both commutative and associative, hence $(\mathrm{P}(\mathrm{A}),\cap )$ is both commutative and associative.

j.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $(\mathrm{P}(\mathrm{A})-\mathrm{\varphi },-)$

Calculation:

Consider the power set of any set $\mathrm{P}(\mathrm{A})$

Consider two elements from the given set.

The subtraction of two same element will give $\mathrm{\varphi }$ , which does not belong to the set.

Therefore, $(\mathrm{P}(\mathrm{A})-\mathrm{\varphi },-)$ is not closed under intersection.

Hence $(\mathrm{P}(\mathrm{A})-\mathrm{\varphi },-)$ is not the algebraic structure.

k.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $\{(0,1),\cdot \}$

Calculation:

Consider the set of $\mathrm{A}=\{0,1\}$

Since, $\begin{array}{l}0\cdot 0=0\cdot 1=1\cdot 0=0\in \mathrm{A}\\ 1\cdot 1=1\in \mathrm{A}\end{array}$

Therefore, $\{(0,1),\cdot \}$ is closed under multiplication.

Hence $\{(0,1),\cdot \}$ is the algebraic structure.

Associative part and commutative part:

It is known that the operation $\cdot$ is both commutative and associative, hence $\{(0,1),\cdot \}$ is both commutative and associative.

l.

To determine

To find which of the following are algebraic structure also find out is the operation commutative or associative.

Explanation of Solution

Given:

  $\{(0,1),+\}$

Calculation:

Consider the set of $\mathrm{A}=\{0,1\}$

Since, $1+1=2\notin \mathrm{A}$

Therefore, $\{(0,1),+\}$ is not closed under addition.

Hence $\{(0,1),+\}$ is not the algebraic structure.