Binary Operation

**novice** · August 4th 2010, 09:17 AM

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 $e*a=e*b=e$ ?

**undefined** · August 4th 2010, 09:51 AM

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.

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.

**novice** · August 4th 2010, 10:24 AM

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.

**undefined** · August 4th 2010, 10:31 AM

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.

**novice** · August 4th 2010, 10:39 AM

Thank you sir. All is clear now.

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