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[ "stage:draft", "article:topic", "authorname:thangarajahp", "license:ccbyncsa", "showtoc:no" ]
Mathematics LibreTexts

1 E Exercises

  • Page ID
    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,
    2.   is commutative,
    3.   is associative, and
    4.   has an 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,
    2.   is commutative,
    3.   is associative, and
    4.   has an 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 \(\PageIndex{8}\):

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

    Exercise \(\PageIndex{9}\):

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

    Exercise \(\PageIndex{10}\):

    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 \(\PageIndex{11}\)

    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 \( \mathbb Q^c\)?
    Answer

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