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1.1: Binary operations

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Binary operation

Definition: Binary operation 

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

Example 1.1.1:

The following are binary operations on Z:

  1. The arithmetic operations, addition +, subtraction , multiplication ×, and division ÷.
  2. Define an operation oplus on Z by ab=ab+a+b,a,bZ.
  3. Define an operation ominus on Z by ab=ab+ab,a,bZ.
  4. Define an operation otimes on Z by ab=(a+b)(a+b),a,bZ.
  5. Define an operation oslash on Z by ab=(a+b)(ab),a,bZ.
  6. Define an operation min on Z by ab=min.
  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.

 


This page titled 1.1: Binary operations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.

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