1.1 Binary operations

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 \( a^2=a,\forall a \in \mathbb{Z}.\)

This is a contradiction.

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.