Answer
464.7k+ views
Hint: Perform the given binary operation on the elements to check the condition of associativity and commutativity. Commutative property holds when for any two elements a and b, we have \[a*b=b*a\]. Associative property holds when for any three elements a, b and c, we have \[a*\left( b*c \right)=\left( a*b \right)*c\].
Complete step-by-step answer:
We have a binary operation \['*'\] defined on set of integers such that \[a*b=a+b+ab\] holds for all \[a,b\in \mathbb{Z}\]. We have to check the associativity and commutativity on this operation.
Commutative property holds when for any two elements a and b, we have \[a*b=b*a\].
We know that \[a*b=a+b+ab\].
We will evaluate the value of \[b*a\]. We have \[b*a=b+a+ba\].
We know that for any two integers a and b, commutativity holds for the operations addition and multiplication. Thus, we have \[a+b=b+a\] and \[ab=ba\].
So, we have \[b*a=b+a+ba=a+b+ab=a*b\].
Thus, commutative property holds for the binary relation \['*'\].
We will now check the associative property.
We will evaluate the value of \[a*\left( b*c \right)\] and \[\left( a*b \right)*c\].
We have \[\left( a*b \right)*c=(a+b+ab)*c=\left( a+b+ab \right)+c+\left( a+b+ab \right)c\].
Further simplifying the above equation, we have \[\left( a*b \right)*c=a+b+ab+c+ac+bc+abc\].
Thus, we have \[\left( a*b \right)*c=a+b+c+ac+bc+ab+abc\].
We now have \[a*\left( b*c \right)=a*\left( b+c+bc \right)=a+\left( b+c+bc \right)+a\left( b+c+bc \right)\].
Further simplifying the equation, we have \[a*\left( b*c \right)=a+b+c+bc+ab+ac+abc\].
Thus, we have \[a*\left( b*c \right)=a+b+c+ab+ac+bc+abc\].
We observe that \[a*\left( b*c \right)=\left( a*b \right)*c\].
Thus, the associative property holds as well.
Hence, commutativity and associativity both hold for the binary operation \['*'\].
Note: One must clearly know the definition of associativity and commutativity. Also, it’s necessary to know that addition and multiplication are associative and commutative on the set of integers. We need to use this property of addition and multiplication to check the associativity and commutativity of the given binary operation.
Complete step-by-step answer:
We have a binary operation \['*'\] defined on set of integers such that \[a*b=a+b+ab\] holds for all \[a,b\in \mathbb{Z}\]. We have to check the associativity and commutativity on this operation.
Commutative property holds when for any two elements a and b, we have \[a*b=b*a\].
We know that \[a*b=a+b+ab\].
We will evaluate the value of \[b*a\]. We have \[b*a=b+a+ba\].
We know that for any two integers a and b, commutativity holds for the operations addition and multiplication. Thus, we have \[a+b=b+a\] and \[ab=ba\].
So, we have \[b*a=b+a+ba=a+b+ab=a*b\].
Thus, commutative property holds for the binary relation \['*'\].
We will now check the associative property.
We will evaluate the value of \[a*\left( b*c \right)\] and \[\left( a*b \right)*c\].
We have \[\left( a*b \right)*c=(a+b+ab)*c=\left( a+b+ab \right)+c+\left( a+b+ab \right)c\].
Further simplifying the above equation, we have \[\left( a*b \right)*c=a+b+ab+c+ac+bc+abc\].
Thus, we have \[\left( a*b \right)*c=a+b+c+ac+bc+ab+abc\].
We now have \[a*\left( b*c \right)=a*\left( b+c+bc \right)=a+\left( b+c+bc \right)+a\left( b+c+bc \right)\].
Further simplifying the equation, we have \[a*\left( b*c \right)=a+b+c+bc+ab+ac+abc\].
Thus, we have \[a*\left( b*c \right)=a+b+c+ab+ac+bc+abc\].
We observe that \[a*\left( b*c \right)=\left( a*b \right)*c\].
Thus, the associative property holds as well.
Hence, commutativity and associativity both hold for the binary operation \['*'\].
Note: One must clearly know the definition of associativity and commutativity. Also, it’s necessary to know that addition and multiplication are associative and commutative on the set of integers. We need to use this property of addition and multiplication to check the associativity and commutativity of the given binary operation.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)