Identity Element of Binary Operations

Therefore, the binary operation does not have an identity element.In summary, the conversation discusses determining whether a given operation has an identity element, and the attempted solution involves finding a value for the identity element based on the given operation. However, it is concluded that the operation does not have an identity element, as the value found (1/3) is not an integer and therefore not in the set in which the operation is defined.
  • #1
gbean
43
0

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!
 
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  • #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
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.
 

What is the identity element of a binary operation?

The identity element of a binary operation is the element that when combined with any other element in the operation, returns the same element. In other words, it is the element that acts as the "neutral" element in the operation.

Why is the identity element important in binary operations?

The identity element is important because it allows us to perform operations without changing the value of the other element. It is also necessary for the existence of inverses in operations.

How do you find the identity element of a binary operation?

To find the identity element, you need to perform the operation with every element in the set. The element that returns the same value is the identity element.

Is the identity element unique in a binary operation?

Yes, the identity element is unique in a binary operation. This means that for a given operation, there can only be one identity element.

What happens if a binary operation does not have an identity element?

If a binary operation does not have an identity element, it is not considered a valid operation. This is because the identity element is a necessary component for performing operations and finding inverses.

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