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# 1 E Exercises

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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.

#### 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.