# 1.1: Binary operations

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## 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 $$\PageIndex{1}$$:

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

1. The arithmetic operations, 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 explore the binary operations, before we proceed:

Example $$\PageIndex{2}$$:

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. $$2 \wedge 3 =3$$.

Exercise $$\PageIndex{2}$$

1. $$-2 \oplus 3$$.
2. $$-2 \otimes 3$$.
3. $$-2 \oslash 3$$.
4. $$-2 \ominus 3$$.
5. $$-2 \vee 3$$.
6. $$-2 \wedge 3$$.

$$-5, 1,-5,-2,3$$

### Properties:

#### Closure property

##### Definition: Closure

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 $$\PageIndex{3}$$: 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 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 $$a \ast_3 b = a+b-3, \forall a,b \in\mathbb{Z}$$.

Exercise $$\PageIndex{1}$$

Determine whether the operation ominus on $$\mathbb{Z_+}$$ is closed?

The operation ominus on $$\mathbb{Z_+}$$ is closed.

Example $$\PageIndex{4}$$: Counter Example

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

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

##### Summary of arithmetic operations and corresponding sets:
 $$+$$ $$\times$$ $$-$$ $$\div$$ $$\mathbb{Z_+}$$ closed closed not closed not closed $$\mathbb{Z}$$ closed closed closed not closed $$\mathbb{Q}$$ closed closed closed closed (only when $$0$$ is not included) $$\mathbb{R}$$ closed closed closed closed (only when $$0$$ is not included)

#### Associative property

##### Definition: Associative

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 on $$\mathbb{Z_+}$$. (You don't need to prove them!).

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

Example $$\PageIndex{5}$$: 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 $$\PageIndex{6}$$: 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 $$\mathbb{Z}$$.

#### Commutative property

##### 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 on $$\mathbb{Z_+}$$. (You don't need to prove them!).

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

Example $$\PageIndex{7}$$: 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 $$\PageIndex{8}$$: 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 $$\mathbb{Z}$$. $$\Box$$

#### Identity

##### 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 $$\PageIndex{9}$$: 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 $$\PageIndex{10}$$: Identity

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

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 $$\PageIndex{11}$$:

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

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 $$a^2=a,\forall a \in \mathbb{Z}.$$

This is a contradiction. Thus $$e=0$$ is not an identity. Hence $$e\ne 0.$$

Choose $$a=1$$. Then $$2e+e^2=0 \imples e(2+e)=0.$$ Since $$e\ne 0$$, $$e=-2$$ This will not work for $$a=0.$$

For any other values of $$e$$ will not work $$a=0$$.

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

#### Distributive Property

##### 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, $$4(10+6)=(4)(10)+(4)(6)=40+24=64$$.

Further, we extend to $$(a+b)(c+d) =ac+ad+bc+bd$$ (FOIL).

F-First

O-Outer

I-Inner

L-Last

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

Example $$\PageIndex{12}$$: Find $$(26)(27)$$

$$20$$ $$6$$
$$20$$ $$400$$ $$120$$
$$7$$ $$140$$ $$42$$

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

Let's play a game!

Example $$\PageIndex{13}$$:

Does multiplication distribute over subtraction?

Example $$\PageIndex{14}$$:

Does division distribute over addition ?

Counter Example:

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

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

Example $$\PageIndex{15}$$:

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

Counter Example:

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

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 $$441 \ne 961,$$  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.

This page titled 1.1: Binary operations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.