Question

# Check the commutativity and associativity of the following binary operations: $'*'$ on $\mathbb{Z}$ defined by $a*b=a+b+ab$ for all $a,b\in \mathbb{Z}$.

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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 $a*\left( b*c \right)=\left( a*b \right)*c$.

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.