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# 1 E Exercises

• • Contributed by Pamini Thangarajah
• Associate Professor (Mathematics & Computing) at Mount Royal University

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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 $$\mathbb Q^c$$?