Question

# Let o be a binary operation on the set ${{Q}_{0}}$ of all non-zero rational numbers defined by $aob=\dfrac{ab}{2}$, for all $a,b\in {{Q}_{0}}$. Show that operation $'o'$ is commutative.

Hint: In the question, we want to show that $'o'$ is commutative. So write the binary operations and use the commutative property. You will get the answer to try it.

So in question, it is mentioned that we have to check whether $'o'$ is commutative.
It is given in the question that $aob=\dfrac{ab}{2}$.
A binary operation is simply a rule for combining two values to create a new value. The most widely known binary operations are those learned in elementary school: addition, subtraction, multiplication, and division on various sets of numbers.
It is possible to define "new" binary operations.
Sometimes, a binary operation on a finite set (a set with a limited number of elements) is displayed in a table that shows how the operation is to be performed. Binary operations on a set are calculations that combine two elements of the set (called operands) to produce another element of the same set.
The binary operations on a non-empty set A are functions from $A\times A$ to $A$. The binary operation: $A\times A$ to A. It is an operation of two elements of the set whose domains and co-domain are in the same set.
An operation of arity (arity is the number of arguments or operands taken by a function or operator.) two that involve several sets are sometimes also called a binary operation. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and the scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions.
So commutative property is the one that refers to moving stuff around. In addition, the rule is $a+b=b+a$; in numbers, this means $2+3=3+2$.
For multiplication, the rule is $ab=ba$; in numbers, this means $2\times 3=3\times 2$. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.
So $aob=\dfrac{ab}{2}$
So let us check for $boa=\dfrac{ba}{2}$
So we get,
$aob=boa$
So we can see that commutative property is proved.

Note: Read the question carefully. Donâ€™t jumble yourself with the operations. Also, you should know the commutative and associative properties. Donâ€™t get confused about the concepts of properties. You should be familiar with binary operations.