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Identity Element of Binary Operations

  • Thread starter gbean
  • Start date
  • #1
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Homework Statement


Determine whether the operation has an identity element.
x*y = 3xy


Homework Equations


e*x = x*e = x, if this holds, e is an identity element


The Attempt at a Solution


My attempt:
x*z = z*x = 3xz, then 3xz = x <=> z = 1/3 => e = 1/3.

But the answer key in the back of the book says that this binary operation has no identity element. Am I wrong, or is the book? Thanks!
 

Answers and Replies

  • #2
It depends on what the relation is defined on. Are x &y integers or real numbers ? If x and y are integers then it is clear that 1/3 is not.
 
  • #3
43
0
Yes, they are integers. Why is this obvious?
 
  • #4
The identity has to be in the group/set . 1/3 is not an integer hence, it is not in the set in which the binary operation is defined.
 

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