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Mathematics LibreTexts

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,
  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.

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.