1.E: Exercises

Exercise \(\PageIndex{1}\): Binary operations

Evaluate the following:

1. \(3 \oplus 4\)
2. \(3 \ominus 4\)
3. \(3\oslash 4\)
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,

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 \(\mathbb{Z}\),

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,

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 \(\mathbb{Z}\),

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.

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 \(S\),

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 \( \mathbb Q^c\)?