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Talk:Majoring in Mathematics (view source)
Revision as of 03:22, February 28, 2012
, 03:22, February 28, 2012→Homomorphism
A homomorphism is a function/relation that maps one algebraic structure to another and preserves its structure. It does not equate them, so I removed it because its definition was erroneous. A comparison illustrates why the original definition is incorrect. If <math>f(x) = x+2</math>, then <math>f(3) = 5</math>. This is a relation that maps <math>\mathbb{R}</math> to <math>\mathbb{R} + 2</math>. It does not, however, equate them. Even though <math>f(x)</math> maps 3 to 5, it is not stating that <math> 3 = 5</math>. This is a simple example, but hopefully it illustrates the proper definition. Thank you! [[User:KevinDavis|Kevin Davis]] <sup>[[User talk:KevinDavis|Talk]]</sup> 09:09, 27 February 2012 (EST)
A homomorphism is a function/relation that maps one algebraic structure to another and preserves its structure. It does not equate them, so I removed it because its definition was erroneous. A comparison illustrates why the original definition is incorrect. If <math>f(x) = x+2</math>, then <math>f(3) = 5</math>. This is a relation that maps <math>\mathbb{R}</math> to <math>\mathbb{R} + 2</math>. It does not, however, equate them. Even though <math>f(x)</math> maps 3 to 5, it is not stating that <math> 3 = 5</math>. This is a simple example, but hopefully it illustrates the proper definition. Thank you! [[User:KevinDavis|Kevin Davis]] <sup>[[User talk:KevinDavis|Talk]]</sup> 09:09, 27 February 2012 (EST)
:This is incorrect, the map you've described is not a homomorphism. A homomorphism <math>\phi:A\rightarrow B</math> satisfies, for all a,b in A, <math>\phi(ab)=\phi(a)\phi(b).</math> I am not talking about claiming that 3=5, I am talking about claiming that the Klein 4-group is the same as Z_2xZ_2.--[[User:Bogart12]]
:This is incorrect, the map you've described is not a homomorphism. A homomorphism <math>\phi:A\rightarrow B</math> satisfies, for all a,b in A, <math>\phi(ab)=\phi(a)\phi(b).</math> I am not talking about claiming that 3=5, I am talking about claiming that the Klein 4-group is the same as Z_2xZ_2.--[[User:Bogart12]]
::What precisely do you mean when you say that the "Klein 4-group is the ''same'' as Z_2xZ_2." As I've mentioned before, you need to be careful when you use such words as "same" or "equal." If all you mean is there exists a homomorphism between the two groups, well then that's tautological and completely uninteresting. If you mean something stronger, well what then be more specific so we can determine if such a claim is true or rather the result of liberal bias. --[[User:JustinD|JustinD]] 22:22, 27 February 2012 (EST)