# Show that S can have at most one object

Let S be a set having an operation ∗ which assigns an element a x b of S for any a, b ∈ S . Let us assume that the following two rules hold:

If a, b are any objects in S, then a x b = a . If a, b are any objects in S, then a x b = b x a .

With this, it is a factor that S has more than one object .

Lets assume x and y are two objects of S

IF WE USE THE RULE a*b=a then x= x*y

Using the rule a*b = b*a then x*y = y*x

With this a*b = a hence in our scenario y*x = y

Based on this illustration it is determined that x is equals to y

Based on our assumption, this is a contradiction since x and y are two separate entities of distinction.

So our findings are wrong since the two objects that belong to S as an entity are not distinct.

On this provision, it is evident that S can have at most one object

final answer

With this, it is a factor that S has more than one object .