
# 1.E: Exercises


## Exercise $$\PageIndex{1}$$: Binary operations

Evaluate the following:

1. $$3 \oplus 4$$
2. $$3 \ominus 4$$
3. $$3\odot 4$$
4. $$3 \otimes 4$$

## Exercise $$\PageIndex{2}$$: Ominus

For $$a, b \in \mathbb{Z},$$ define an operation $$\ominus$$, by $$a \ominus b= ab+a-b.$$ Determine whether $$\ominus$$ on $$\mathbb{Z}$$,

1. is closed,

Proof:

Let $$a, b \in \mathbb{Z}.$$

Then consider$$a \ominus b= ab+a-b.$$

Since ·, + and - are closed on $$\mathbb{Z}$$, $$ab, (ab)+a,$$ and $$(ab+a)-b$$ are in  $$\mathbb{Z}$$.

Hence $$a \ominus b \in \mathbb{Z}.$$

Thus, the binary operation is closed on $$\mathbb{Z}$$. ⬜

2. is commutative,

Counterexample:

Choose $$a=3$$ and $$b=4.$$

Then consider $$3 \ominus 4=(3)(4)+3-4 =12+3-4 =15-4=11$$.

Now consider $$4 \ominus 3=(4)(3)+4-3=12+4-3 =16-3 =13.$$

Since $$11 \ne 13,$$ the binary operation $$\ominus$$ is not commutative on $$\mathbb{Z}$$.

3. is associative,

Counterexample:

Choose $$a=3, b=4, c=2.$$

Then consider $$(3 \ominus 4) \ominus 2=[(3)(4)+3-4] \ominus 2 =11 \ominus 2 =(11)(2)+11-2 =22+11-2 =31.$$

Now consider, $$3 \ominus (4 \ominus 2)=3 \ominus [(4)(2)+4-2]=3 \ominus 10 =(3)(10)+3-10 =30+3-10 =23.$$

Since  $$31 \ne 23,$$  the binary operation is not associative on $$\mathbb{Z}$$.

4. has an identity.

Let  $$e$$ be the identity on $$\mathbb{Z}$$. Then $$a \ominus e=e \ominus a=a, a∈\mathbb{Z}$$.

Now consider, $$a \ominus e=e \ominus a.$$

$$ae+a-e=ea+e-a.$$

$$ae-ea+2a=2e.$$

$$ae=ea$$ since · is commutative on $$\mathbb{Z}$$.

Thus,  $$e=a.$$

Now consider $$ae=ae+a-e=a.$$

If$$e=a,$$ then $$a^2=a, a∈ \mathbb{Z}$$.

This is a contradiction.

Thus, $$(\mathbb{Z}, \ominus )$$ has no identity.

## Exercise $$\PageIndex{3}$$: Oplus

For $$a, b \in \mathbb{Z},$$ define an operation $$\oplus$$, by $$a \oplus b= ab+a+b.$$ Determine whether $$\oplus$$ on $$\mathbb{Z}$$,

1. is closed,
2. is commutative,
3. is associative, and
4. has an identity.

What happens to the result for the above questions, if we change the set to $$\mathbb{Z} \setminus \{-1\}$$?

## Exercise $$\PageIndex{4}$$: Oslash

For $$a, b \in \mathbb{Z},$$ define an operation $$\oslash$$, by $$a \oslash b= (a+b)(a-b).$$ Determine whether $$\oslash$$ on $$\mathbb{Z}$$,

1. is closed,
2. is commutative,
3. is associative, and
4. has an identity.

## Exercise $$\PageIndex{5}$$: Otimes

For $$a, b \in \mathbb{Z},$$ define an operation $$\otimes$$, by $$a \otimes b= (a+b)(a+b).$$ Determine whether $$\otimes$$ on $$\mathbb{Z}$$,

1. is closed,

Proof:

Let $$a, b \in \mathbb{Z}$$. Then consider, $$a \otimes b= (a+b)(a+b).$$

since · is commutative on $$\mathbb{Z}$$, $$(a+b)(a+b)=a^2+2ab+b^2$$. Since  ·, + are closed on $$\mathbb{Z}$$, $$a^2, 2ab, b^2, a^2+2ab, and (a^2+2ab)+b^ 2 \in \mathbb{Z}$$.

Hence, $$a^2+2ab+b^ 2 \in \mathbb{Z}$$. Thus  $$a \otimes b \in \mathbb{Z}$$.

Therefore, the binary operation $$\otimes$$ is closed on $$\mathbb{Z}$$.⬜

2. is commutative,

Proof:

Let $$a, b \in \mathbb{Z}$$. Then consider, $$a \otimes b= (a+b)(a+b) =a^2+2ab+b^2= b^2+2ba+a^2 = (b+a)(b+a)= b \otimes a .$$

Therefore, the binary operation $$\otimes$$ is commutative on $$\mathbb{Z}$$.⬜

3. is associative,

Counterexample:

Choose $$a=2, b=3, c=4.$$

Then consider, $$( a \otimes b ) \otimes c​​​=[(2 + 3)(2 + 3)] \otimes 4= 25 \otimes 4= ​​​​(25 + 4)(25 + 4)=841$$.

Now consider $$a \otimes (b \otimes c)=2 \otimes (3 + 4)(3+ 4)= 2 \otimes 49 =(2 +49)(2 + 49) =2601.$$

Since $$841 \ne 2601$$, the binary operation $$\otimes$$   is not associative on $$\mathbb{Z}$$. ⬜

4. has an identity.

Let $$e$$ be the identity on $$( \mathbb{Z}, \otimes$$.

Then consider, $$a \otimes e=e \otimes a=a, \forall \mathbb{Z}.$$

Now, $$a = e \otimes a = (e+a)(e+a)= e(e+a)+a(e+a)$$, since · is associative on $$\mathbb{Z}.$$

$$a = e^2+ea+ae+a^2=e^2+2ae+a^2. Then choose a=0. Thus, \(e^2 = 0$$, which implies $$e = 0.$$

Hence $$a^ 2 = a, \forall \mathbb{Z}.$$ This is a contradiction.

Thus,$$( \mathbb{Z}, \otimes$$ has no identity. ⬜

## Exercise $$\PageIndex{6}$$: Max

For $$a, b \in \mathbb{Z},$$ define an operation $$\land$$, by $$a\land b= max \{a,b\}.$$ Determine whether $$\land$$ on $$\mathbb{Z}$$,

1. is closed,
2. is commutative,
3. is associative, and
4. has an identity.

## Exercise $$\PageIndex{7}$$: Min

For $$a, b \in \mathbb{Z},$$ define an operation $$\lor$$, by $$a\lor b= min \{a,b\}.$$ Determine whether $$\lor$$ on $$\mathbb{Z}$$,

1. is closed,
2. is commutative,
3. is associative, and
4. has an identity.

## Exercise $$\PageIndex{8}$$: Defection

For $$a, b \in \mathbb{Z},$$ define an operation $$\star_5$$, by $$a\star_5 b= a+b-5 .$$ Determine whether $$\star_5$$ on $$\mathbb{Z}$$,

1. is closed,
2. is commutative,
3. is associative, and
4. has an identity.

is closed, is commutative, is associative, and has an identity.

## Exercise $$\PageIndex{9}$$: Distributive

Determine whether $$\otimes$$ is distributive over $$\oplus$$ on $$\mathbb Z$$.

Counterexample:

Choose $$a = 2, b=3, c=4.$$

Then consider $$2 \otimes (3 \oplus 4)=2 \otimes [(3)(4) + 3 + 4] =2 \otimes 19=(2 + 19)(2 + 19) =441.$$

Now consider $$(2 \otimes 3) \oplus (2 \otimes 4)=[(2 + 3)(2 + 3)] \oplus [(2 + 4)(2 + 4)]= 25 \oplus 36 =(25)(36)+25+36 =961.$$

Since  $$441 \ne 961,$$  the binary operation $$\otimes$$ is not distributive  $$\oplus$$ overon $$\mathbb Z$$. ⬜

## Exercise $$\PageIndex{10}$$: Distributive

Determine whether $$\oslash$$ is distributive over $$\oplus$$ on $$\mathbb Z$$.

## Exercise $$\PageIndex{11}$$: Even Multiplication

Let $$S=\{1, 2,4,6, \dots \}$$, that is $$S$$ is the set of all even positive integers and $$1$$. Use the multiplication $$\times$$ as the binary operation.

Determine whether the multiplication $$\times$$ on $$S$$,

1. is closed,
2. is commutative,
3. is associative, and
4. has an identity.

## Exercise $$\PageIndex{12}$$: Arithmetic operations on Rationals and Irrationals

1. Is addition closed on $$\mathbb Q$$?
2. Is multiplication closed on $$\mathbb Q$$?
3. Is addition closed on $$\mathbb Q^c$$?
4. Is multiplication closed on $$\mathbb Q^c$$?