1.1: Binary operations

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. $-2 \wedge 3$.

Answer

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

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?

Answer

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 $\mathbb{Z}$.

Counterexample:

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 $\mathbb{Z}$.

Counterexample:

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?

Answer

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?

Answer

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?

Answer

Counterexample:

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 $\mathbb{Z}$ ?

Answer

Counterexample:

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.