Question

Is $*$ defined on the set $\\{ 1,2,3,4,5\\}$ by $a * b = LCM{\text{ of }}a{\text{ and }}b$, a binary operation? Justify your answer.

Answer 1

The set consists of five elements. For $*$ be a binary operation, it must satisfy the closure property for every pair of elements. That is, every pair of numbers that have their LCM belongs to the set.

Formula used:
An operation $*$ defined on a set $A$ is said to be a binary operation if for every $a,b \in A$, $a * b \in A$.

Complete step-by-step answer:
It is asked that $*$ defined on the set $\\{ 1,2,3,4,5\\}$ by $a * b = LCM{\text{ of }}a{\text{ and }}b$ is a binary operation or not.
An operation $*$ defined on a set $A$ is said to be a binary operation if for every $a,b \in A$, $a * b \in A$.
To prove something, we have to establish it generally.
But to disprove something we just need a counter example.
That is, an example lacking this property will be enough.
So if there exists a pair $a,b \in \\{ 1,2,3,4,5\\}$ such that their LCM is not present in $\\{ 1,2,3,4,5\\}$, then we can say that it is not a binary operation.
Consider the numbers $2$ and $3$ belong to the given set.
We know that the least common multiple of these numbers is $6$.
But we can see that it does not belong to the given set.
So we get a counter example to establish that this is not a binary relation.

Therefore, $*$ defined on the set $\\{ 1,2,3,4,5\\}$ by $a * b = LCM{\text{ of }}a{\text{ and }}b$ is not a binary operation.

Note:
To disprove here we considered the pair $2,3$. But this is not unique. Also we have numbers $2$ and $5$ with LCM outside the set. But one example is enough.