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Mathematics LibreTexts

1.2 Binary operations

[ "stage:draft", "article:topic", "authorname:thangarajahp", "license:ccbyncsa", "showtoc:yes" ]
  • Page ID
    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. \(2 \wedge 3 =3\).

    Exercise \(\PageIndex{1}\)

    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:

     

    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.