I was reading about Composition table or Cayley Table; one of the points my book presents is that
If all the entries of the table are elements of set $S$ and each element of $S$ appears once and only once in each row and column, then the operation is a binary operation.
The first row of the table contains elements $a_1*a_1,\,a_1*a_2,\,a_1*a_3,\,\cdots$
Now, according to the statement, in order for $*$ to be a binary operation, $a_1*a_1\ne \,a_1*a_2\ne \,a_1*a_3\ne \,\cdots\,$ is it so?
If yes, I'm not getting the reason behind it. What would be the problem if $a_1*a_1= \,a_1*a_2 = \,a_1*a_3\;?$
Can anyone please explain to me what the bold statement means?