# Show that S can have at most one object

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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 .

Thad Answered question September 16, 2022

### 2 Answers

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

Thad Answered question September 16, 2022
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Arnica (anonymous) 0 Comments

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 .

Teacher Changed status to publish August 26, 2022