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Mathematics LibreTexts

1.1 Binary operations

  • Page ID
    7419
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    Think out Loud

    What is the meaning of a unary operation?

    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 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 understand 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 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 \(\PageIndex{3}\): 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
    \(\mathbb{R}\) closed closed closed closed

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

    Below is an example of a 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 \(\mathbb{Z}\).

    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 \(\mathbb{Z}\). \( \Box\)

    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.

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

    Further, \( (a+b)(c+d)\)

    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, 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 \(\PageIndex{15}\):

    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.