# Binary Operation

• Aug 4th 2010, 09:17 AM
novice
Binary Operation
A binary operation $*$ is defined on the set $S=\{a,b,c\}$ by $x*y=x$ for all $x,y\inS$. Determine which of the properties (Associative, Identity, Inverse, Commutative) are satisfied by $(S,*).$

Solution:
Let $x,y$, and $z$ be any three elements of $S$ (distinct of not). Then $x*(y*z) = x*y=x$, while $(x*y)*z=x*z=x$. Thus (S,*) is associative. Now (S,*) has no identity since for every element $e \in S$, it follows that $e*a=e*b=e$ and so it is impossible for $e*a=a$ and $e*b=b$. Since (S,*) has no identity, the question of inverses does not apply here. Certainly, $(S,*)$ is not commutative since $a*b=a$ while $b*a=b$.

Question:
I suppose $e$ is an identity element. Since (S,*) has no identity why it contradict itself by saying "(S,*) has no identity since for every element " $e\in S$"?

$e*a=a$ and $e*b=b$ make sense, but what is $e*a=e*b=e$?
• Aug 4th 2010, 09:51 AM
undefined
Quote:

Originally Posted by novice
A binary operation $*$ is defined on the set $S=\{a,b,c\}$ by $x*y=x$ for all $x,y\inS$. Determine which of the properties (Associative, Identity, Inverse, Commutative) are satisfied by $(S,*).$

Solution:
Let $x,y$, and $z$ be any three elements of $S$ (distinct of not). Then $x*(y*z) = x*y=x$, while $(x*y)*z=x*z=x$. Thus (S,*) is associative. Now (S,*) has no identity since for every element $e \in S$, it follows that $e*a=e*b=e$ and so it is impossible for $e*a=a$ and $e*b=b$. Since (S,*) has no identity, the question of inverses does not apply here. Certainly, $(S,*)$ is not commutative since $a*b=a$ while $b*a=b$.

Question:
I suppose $e$ is an identity element. Since (S,*) has no identity why it contradict itself by saying "(S,*) has no identity since for every element " $e\in S$"?

$e$ is not an identity element, it is just a letter. You can think of it as "candidate for identity element." The text says "for every element $e \in S$", not "for an identity element $e \in S$". The text does not contradict itself.

Quote:

Originally Posted by novice
$e*a=a$ and $e*b=b$ make sense, but what is $e*a=e*b=e$?

Look at the definition of identity element. The logic here is for left identity element in particular. The point is that if $e$ is an identity element then $e*a = a$ and $e*b = b$, therefore $e*a\ne e*b$. This produces a contradiction since we know $e*a=e*b=e$. Therefore, no matter what choice of $e$ we make, $e$ is not an identity element.
• Aug 4th 2010, 10:24 AM
novice
Quote:

Originally Posted by undefined
$e$ is not an identity element, it is just a letter. You can think of it as "candidate for identity element

If $e$ is only a possible candidate, the what operational property is there that can produce $e*a=e*b=e$, or is this just non-sense created for making the point.
• Aug 4th 2010, 10:31 AM
undefined
Quote:

Originally Posted by novice
If $e$ is only a possible candidate, the what operational property is there that can produce $e*a=e*b=e$, or is this just non-sense created for making the point.

$e$ is an element of $S$. Direct application of the definition of $*$ gives $e*a=e*b=e$, regardless of what $e$ is. It is not nonsense.
• Aug 4th 2010, 10:39 AM
novice
Thank you sir. All is clear now.