Identity Element of Binary Operations

Click For Summary

Homework Help Overview

The discussion revolves around determining whether a specific binary operation, defined as x*y = 3xy, possesses an identity element. The original poster attempts to find an identity element based on the condition e*x = x*e = x.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster calculates a potential identity element and questions the discrepancy between their result and the answer key. Other participants inquire about the nature of the elements involved (integers vs. real numbers) and clarify that the identity must belong to the same set as the operands.

Discussion Status

The discussion is exploring the implications of the set in which the operation is defined, with some participants suggesting that the identity element must be an integer, while the original poster's proposed identity element is not. There is an ongoing examination of the definitions and constraints related to the operation.

Contextual Notes

Participants note that the identity element must be within the same set as the operands, which in this case are integers. The original poster's proposed identity element of 1/3 is questioned based on this requirement.

gbean
Messages
40
Reaction score
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!
 
Physics news on Phys.org
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.
 
Yes, they are integers. Why is this obvious?
 
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.
 

Similar threads

Replies
2
Views
2K
Replies
3
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 4 ·
Replies
4
Views
3K
  • · Replies 1 ·
Replies
1
Views
1K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 1 ·
Replies
1
Views
1K
  • · Replies 3 ·
Replies
3
Views
2K