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Mathematics LibreTexts

1.E: Exercises

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    7421
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    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,
    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 \( \mathbb Q^c\)?