
# 1.1 Binary operations

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7419
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Thinking out loud

## Binary operation

#### Definition: Binary operation

Let $$S$$ be a  non-empty set,  and $$\star$$ said to be a binary operation on $$S$$ , if $$a \star b$$ is defined for all  $$a,b \in S$$.    In other words, $$\star$$ is a rule for any two elements in the set $$S$$.

#### Example $$\PageIndex{1}$$: Binary operations

The following are binary operations on  $$\mathbb{Z}$$:

1.   The addition $$+$$ , subtraction $$-$$, multiplication $$\times$$ , and division  $$\div$$.
2.   Define an operation oplus  on $$\mathbb{Z}$$ by $$a \oplus b =ab+a+b, \forall a,b \in\mathbb{Z}$$.
3.   Define an operation  ominus on $$\mathbb{Z}$$ by $$a \ominus b =ab+a-b, \forall a,b \in\mathbb{Z}$$.
4.   Define an operation  otimes on $$\mathbb{Z}$$ by $$a \otimes b =(a+b)(a+b), \forall a,b \in\mathbb{Z}$$.
5.   Define an operation oslash on $$\mathbb{Z}$$ by $$a \oslash b =(a+b)(a-b), \forall a,b \in\mathbb{Z}$$.
6.   Define an operation min  on $$\mathbb{Z}$$ by $$a \vee b =\min \{a,b\}, \forall a,b \in\mathbb{Z}$$.
7.   Define an operation max  on $$\mathbb{Z}$$ by $$a \wedge b =\max \{a,b\}, \forall a,b \in\mathbb{Z}$$.
8.   Define an operation  defect on $$\mathbb{Z}$$ by $$a \ast_3 b = a+b-3, \forall a,b \in\mathbb{Z}$$.

Lets understand the binary operations, before we proceed:

Example $$\PageIndex{1}$$:

1. $$2 \oplus 3=(2)(3)+2+3=11$$.
2. $$2 \otimes 3=(2+3)(2+3)=25$$ .
3. $$2 \oslash 3=(2+3)(2-3)=-5$$.
4. $$2 \ominus 3=(2)(3)+2-3=5$$.
5. $$2 \vee 3= 2$$.
6. $$2 \wedge 3 =3$$.

### Properties:

#### Definition :  Closure property

Let $$S$$ be a non-empty set.   A binary operation $$\star$$  on $$S$$ is said to be a closed binary operation on $$S$$  , if $$a \star b \in S, \forall a, b \in S$$.

Below we shall give some examples of closed binary operations, that will be further explored in class.

##### Example $$\PageIndex{2}$$:   Closed binary operations

The following are closed binary operations on  $$\mathbb{Z}$$.

1.   The addition $$+$$ , subtraction $$-$$, and multiplication $$\times$$.
2.   Define an operation oplus on $$\mathbb{Z}$$ by $$a \oplus b =ab+a+b, \forall a,b \in\mathbb{Z}$$.
3.   Define an operation  ominus on $$\mathbb{Z}$$ by $$a \ominus b =ab+a-b, \forall a,b \in\mathbb{Z}$$.
4.   Define an operation  otimes on $$\mathbb{Z}$$ by $$a \otimes b =(a+b)(a+b), \forall a,b \in\mathbb{Z}$$.
5.   Define an operation odot on $$\mathbb{Z}$$ by $$a \odot b =(a+b)(a-b), \forall a,b \in\mathbb{Z}$$.
6.   Define an operation min  on $$\mathbb{Z}$$ by $$a \vee b =\min \{a,b\}, \forall a,b \in\mathbb{Z}$$.
7.   Define an operation max  on $$\mathbb{Z}$$ by $$a \wedge b =\max \{a,b\}, \forall a,b \in\mathbb{Z}$$.
8.   Define an operation  defect on $$\mathbb{Z}$$ by $$a \ast_3 b = a+b-3, \forall a,b \in\mathbb{Z}$$.
##### Example $$\PageIndex{3}$$: Counter Example

Division ($$\div$$ ) is not a closed binary operations on  $$\mathbb{Z}$$.

$$2, 3 \in \mathbb{Z}$$ but  $$\frac{2}{3} \notin \mathbb{Z}$$.

#### Definition: Associative property

Let $$S$$ be a subset of $$\mathbb{Z}$$.  A binary operation $$\star$$  on $$S$$ is said to be  associative , if $$(a \star b) \star c = a \star (b \star c) , \forall a, b,c \in S$$.

We shall assume the fact that the addition ($$+$$)  and the multiplication ($$\times$$) are associative. (You don't need to prove them!).

Below is an example of a proof when the statement is True.

##### Example $$\PageIndex{4}$$: Associative

Determine whether the binary operation oplus is associative on $$\mathbb{Z}$$.

We shall show that the binary operation oplus is associative on $$\mathbb{Z}$$.

Proof:

Let $$a,b,c \in \mathbb{Z}$$.

Then consider, $$(a \oplus b) \oplus c = (ab+a+b) \oplus c = (ab+a+b)c+(ab+a+b)+c= (ab)c+ac+bc+ab+a+b+c$$.

On the other hand,  $$a \oplus (b \oplus c)=a \oplus (bc+b+c)= a(bc+b+c)+a+(bc+b+c)=a(bc)+ab+ac+a+bc+b+c.$$

Since multiplication is associative on  $$\mathbb{Z}$$,  $$(a \oplus b) \oplus c =a \oplus (b \oplus c).$$

Thus, the  binary operation oplus is associative on $$\mathbb{Z}$$. $$\Box$$

Below is an example of how to disprove when a statement is False.

##### Example $$\PageIndex{5}$$: Not Associative

Determine whether the binary operation subtraction ($$-$$) is associative on  $$\mathbb{Z}$$.

Answer:  The binary operation  subtraction ($$-$$) is  not associative on $$\mathbb{Z}$$.

Counter Example:

Choose $$a=2,b=3, c=4,$$ then $$(2-3)-4=-1-4=-5$$, but $$2-(3-4)=2-(-1)=2+1=3$$.

Hence the binary operation subtraction ($$-$$) is  not associative on $$\mathbb{Z}$$.

#### Definition: Commutative property

Let $$S$$ be a non-empty set.  A binary operation $$\star$$  on $$S$$ is said to be commutative, if $$a \star b = b \star a,\forall a, b \in S$$.

We shall assume the fact that the addition ($$+$$)  and the multiplication( $$\times$$) are commutative. (You don't need to prove them!).

Below is the proof of subtraction ($$-$$) NOT being commutative.

#### Example $$\PageIndex{6}$$:  NOT Commutative

Determine whether the binary operation subtraction $$-$$ is commutative on $$\mathbb{Z}$$.

Counter Example:

Choose $$a=3$$ and $$b=4$$.

Then $$a-b=3-4=-1$$, and $$b-a= 4-3=1$$.

Hence the binary operation subtraction $$-$$ is not commutative on $$\mathbb{Z}$$.

#### Example $$\PageIndex{7}$$: Commutative

Determine whether the binary operation oplus is commutative on $$\mathbb{Z}$$.

We shall show that the binary operation oplus is commutative on $$\mathbb{Z}$$.

Proof:

Let $$a,b \in \mathbb{Z}$$.

Then consider, $$(a \oplus b) = (ab+a+b).$$

On the other hand,  $$(b \oplus a) = ba+b+a.$$

Since multiplication is associative on $$\mathbb{Z}$$,  $$(a \oplus b) = (b \oplus a).$$

Thus, the binary operation oplus is commutative on $$\mathbb{Z}$$. $$\Box$$

#### Definition: Identity

A non-empty set $$S$$ with binary operation $$\star$$, is said to have an identity $$e \in S$$, if $$e \star a=a\star e=a, \forall a \in S.$$

Note that $$0$$ is called additive identity on $$( \mathbb{Z}, +)$$, and $$1$$ is called multiplicative identity on $$( \mathbb{Z}, \times )$$.

#### Example $$\PageIndex{8}$$: Is identity unique?

Let $$S$$ be a non-empty set and let $$\star$$ be a binary operation on $$S$$.  If $$e_1$$ and $$e_2$$ are two identities in $$(S,\star)$$ , then $$e_1=e_2$$.

Proof:

Suppose that  $$e_1$$ and $$e_2$$ are two identities in  $$(S,\star)$$.

Then  $$e_1=e_1 \star e_2=e_2.$$

Hence  identity is unique. $$\Box$$

#### Example $$\PageIndex{9}$$: Identity

Does $$( \mathbb{Z}, \oplus )$$ have an identity?

Solution:

Let $$e$$ be the identity on $$( \mathbb{Z}, \oplus )$$.

Then  $$e \oplus a=a\oplus e=a, \forall a \in \mathbb{Z}.$$

Thus $$ea+e+a=a$$, and $$ae+a+e=a$$  $$\forall a \in \mathbb{Z}.$$

Since $$ea+e+a=a$$  $$\forall a \in \mathbb{Z},$$ $$ea+e=0 \implies e(a+1)=0$$  $$\forall a \in \mathbb{Z}.$$

Therefore $$e=0$$.

Now  $$0 \oplus a=a\oplus 0=a, \forall a \in \mathbb{Z}.$$

Hence $$0$$ is the identity on  $$( \mathbb{Z}, \oplus )$$.

#### Example $$\PageIndex{10}$$:

Does  $$( \mathbb{Z}, \otimes )$$ have an identity?

Solution:

Let $$e$$  be the identity on $$( \mathbb{Z}, \otimes )$$.

Then $$e \otimes a=a \otimes e=a, \forall a \in \mathbb{Z}.$$

Thus $$(e+a)(e+a)=(a+e)(a+e) =a, \forall a \in \mathbb{Z}.$$

Now, $$(a+e)(a+e) =a,\forall a \in \mathbb{Z}.$$

$$\implies a^2+2ea+e^2=a,\forall a \in \mathbb{Z}.$$

Choose $$a=0$$ then $$e=0$$.

If $$e=0$$ then $$a^2=a,\forall a \in \mathbb{Z}.$$

Hence,  $$( \mathbb{Z}, \otimes )$$ has no  identity.

#### Definition: Distributive property

Let $$S$$ be a non-empty set.  Let $$\star_1$$ and $$\star_2$$ be two different binary operations on $$S$$.

Then $$\star_1$$ is said to be distributive over $$\star_2$$  on $$S$$ if  $$a \star_1 (b \star_2 c)= (a\star_1 b) \star_2 (a \star_1 c), \forall a,b,c,\in S$$.

Note that the multiplication distributes over the addition on $$\mathbb{Z}.$$  That is, $$4(10+6)=(4)(10)+(4)(6)=40+24=64$$.

This property is very useful to find $$(26)(27)$$ as shown below:

20 6
20 400 120
7 140 42

Hence  $$(26)(27) =400+120+140+42=702$$.

#### Example $$\PageIndex{11}$$:

Does multiplication distribute over subtraction?

#### Example $$\PageIndex{12}$$:

Does division distribute over addition ?

Counter Example:

Choose a = 2, b = 3, c = 4.

Then a $$\div$$(b + c) = 2$$\div$$(3+4)

= 2 $$\div$$ 7.

= $$\frac{2}{7}$$.

and (a $$\div$$ b) + (a $$\div$$ c) = $$\frac{2}{3}$$ + $$\frac{2}{4}$$.

= $$\frac{7}{6}$$.

Since $$\frac{2}{7}$$ ≠ $$\frac{7}{6}$$, the binary operation $$\div$$ is not distributive over +.

#### Example $$\PageIndex{13}$$:

Does $$\otimes$$  distribute over $$\oplus$$ on $$\mathbb{Z}$$ ?

Counter Example:

Choose a = 2, b = 3, c = 4.

Then 2$$\otimes$$(3$$\oplus$$4) = 2$$\otimes$$[(3)(4)+3+4]

= 2$$\otimes$$19

= (2+19)(2+19)

= 441

and (2$$\otimes$$3)$$\oplus$$(2$$\otimes$$4)=[(2+3)(2+3)]$$\oplus$$[(2+4)(2+4)]

= 25$$\oplus$$36

= (25)(36)+25+36

= 961.

Since 441 ≠ 961, the binary operation $$\otimes$$ is not distributive over $$\oplus$$ on $$\mathbb{Z}$$.

#### Summary

In this section, we have learned the following for a non-empty set $$S$$:

1.   Binary operation,
2.   Closure property,
3.   Associative  property,
4.   Commutative property,
5.   Distributive property, and
6.   Identity.