1.E: Exercises

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Jan 20, 2022
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- 1.2: Exponents and Cancellation
- 2: Binary relations

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Exercise $\PageIndex{1}$ : Binary operations

Evaluate the following:

$3 \oplus 4$
$3 \ominus 4$
$3\oslash 4$
$3 \otimes 4$

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 $\mathbb{Z}$ ,

is closed,

Answer

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,

Answer

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,

Answer

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.

Answer

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 $\mathbb{Z}$ ,

is closed,
is commutative,
is associative, and
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 $\mathbb{Z}$ ,

is closed,
is commutative,
is associative, and
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 $\mathbb{Z}$ ,

is closed,

Answer

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,

Answer

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,

Answer

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.

Answer

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 $\mathbb{Z}$ ,

is closed,
is commutative,
is associative, and
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 $\mathbb{Z}$ ,

is closed,
is commutative,
is associative, and
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 $\mathbb{Z}$ ,

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

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

Exercise $\PageIndex{9}$ : Distributive

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

Answer

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 $S$ ,

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

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

Is addition closed on $\mathbb Q$ ?
Is multiplication closed on $\mathbb Q$ ?
Is addition closed on $\mathbb Q^c$ ?
Is multiplication closed on $\mathbb Q^c$ ?

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Exercise 1.E.1\PageIndex{1}: Binary operations

Exercise 1.E.2\PageIndex{2}: Ominus

Exercise 1.E.3\PageIndex{3}: Oplus

Exercise 1.E.4\PageIndex{4}: Oslash

Exercise 1.E.5\PageIndex{5}: Otimes

Exercise 1.E.6\PageIndex{6}: Max

Exercise 1.E.7\PageIndex{7}: Min

Exercise 1.E.8\PageIndex{8}: Defection

Exercise 1.E.9\PageIndex{9}: Distributive

Exercise 1.E.10\PageIndex{10}: Distributive

Exercise 1.E.11\PageIndex{11}: Even Multiplication

Exercise 1.E.12\PageIndex{12}: Arithmetic operations on Rationals and Irrationals

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Exercise $\PageIndex{1}$ : Binary operations

Exercise $\PageIndex{2}$ : Ominus

Exercise $\PageIndex{3}$ : Oplus

Exercise $\PageIndex{4}$ : Oslash

Exercise $\PageIndex{5}$ : Otimes

Exercise $\PageIndex{6}$ : Max

Exercise $\PageIndex{7}$ : Min

Exercise $\PageIndex{8}$ : Defection

Exercise $\PageIndex{9}$ : Distributive

Exercise $\PageIndex{10}$ : Distributive

Exercise $\PageIndex{11}$ : Even Multiplication

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