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Math Help - Binary Operation

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by novice View Post
    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 View Post
    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.
    Last edited by undefined; August 4th 2010 at 10:08 AM. Reason: typo
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  3. #3
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    Quote Originally Posted by undefined View Post
    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.
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    MHF Contributor undefined's Avatar
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    Quote Originally Posted by novice View Post
    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.
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    Thank you sir. All is clear now.
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