If g(f(x) is injective I need to determine if f is injective. If it is false, I need to give a counterexample. I was told that using arrow diagrams helps. Can some one help me get started?
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I know g doesn't have to be injective.
I'm think that it does but I am unsure of how to prove this.
So g(f(x)) is 1-1, so g(f( =g(f( . Then this implies f( =f( . Where do I go from here? Would that last statement prove that ? therefore, f is one to one?
Originally Posted by kathrynmath So g(f(x)) is 1-1, so g(f( =g(f( . Then this implies f( =f( . Where do I go from here? Would that last statement prove that ? therefore, f is one to one? hint: is another way of writing (the statement is true) now you need to show that if then
Originally Posted by Jhevon hint: is another way of writing (the statement is true) now you need to show that if then Ok, so, I'm unsure on how to prove that that . This is where I get stuck.
Originally Posted by kathrynmath Ok, so, I'm unsure on how to prove that that . This is where I get stuck. yes, and i told you how to get around it. follow my hint to start you off: assume . then . but that means ...
Originally Posted by Jhevon yes, and i told you how to get around it. follow my hint to start you off: assume . then . but that means ... Ok, so I start off by assumming f( )=f( )? Then . g(f) is injective, so f( .
Originally Posted by kathrynmath Ok, so I start off by assumming f( )=f( )? Then . g(f) is injective, so f( . ...that brings us back where we started. we know , that's what we assumed! you are supposed to apply my hint where i left off
Originally Posted by Jhevon ...that brings us back where we started. we know , that's what we assumed! you are to apply my hint there... Ok, can I just say because and is injective?
Originally Posted by kathrynmath Ok, can I just say because and is injective? assume . then . but that means . since is injective, this means . thus, is injective
Originally Posted by Jhevon assume . then . but that means . since is injective, this means . thus, is injective Ok, that makes sense.
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