# 1 E Exercises

- Page ID
- 7421

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

Evaluate the following:

- \(3 \oplus 4\)
- \(3 \ominus 4\)
- \(3\odot 4\)
- \(3 \otimes 4\)

## 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,
- is commutative,
- is associative, and
- 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 \(\mathbb{Z}\).

- 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,
- is commutative,
- is associative, and
- 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 \(\mathbb{Z}\).

- 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 \(\PageIndex{8}\):

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 \(S\),

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

Exercise \(\PageIndex{11}\)

- Is addition closed on \( \mathbb Q\)?
- Is multiplication closed on \( \mathbb Q\)?
- Is addition closed on \( \mathbb Q^c\)?
- Is multiplication closed on \( \mathbb Q^c\)?

**Answer**-
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