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Sep
24
awarded  Popular Question
Sep
24
comment Prove that $2^n>n^4$ for all $n\geq 17$
It genuinely seems like these types of problems come easy with an inordinate amount of practice, and having knowledge of general techniques. I would have never considered many of those inequalities.
Sep
24
asked Prove that $2^n>n^4$ for all $n\geq 17$
Sep
24
awarded  Custodian
Sep
24
asked Direct proof of the existence of Strong Induction using the Well Ordering Principle
Sep
22
accepted Prove that $(\mathbb{R}\setminus \{0\},\sim):= ab>0$ is transitive.
Sep
22
accepted Determine whether the function $\alpha:A\times B\rightarrow B\times A$ where $\alpha((a,b))=(b,a)$ is injective and/or surjective
Sep
22
accepted Give an example of two functions $\alpha,\beta$ on a set $A$ such that $\alpha\circ\beta=\mathsf{id}_{A}$ but $\beta\circ\alpha\neq\mathsf{id}_{A}$.
Sep
22
accepted Proving that $\alpha:\mathbb{R}\to\mathbb{R}$ where $\alpha(x)=\frac{x^{3}}{x^{2}+1}$ is bijective
Sep
22
comment Proving that $\alpha:\mathbb{R}\to\mathbb{R}$ where $\alpha(x)=\frac{x^{3}}{x^{2}+1}$ is bijective
I'm afraid that goes way beyond my understanding at this level... I was warned by my prof that trying to find an inverse may end up being too messy, so you've just proven that for me...
Sep
22
comment Proving that $\alpha:\mathbb{R}\to\mathbb{R}$ where $\alpha(x)=\frac{x^{3}}{x^{2}+1}$ is bijective
I don't follow how you converted $x^2+xy+y^2$ as $\frac{3}{4}(x+y)^2+\frac{1}{4}(x-y)^2$
Sep
22
comment Proving that $\alpha:\mathbb{R}\to\mathbb{R}$ where $\alpha(x)=\frac{x^{3}}{x^{2}+1}$ is bijective
This seems very strange to me, claiming that it is injective under the constraint that $x=y=0$. Shouldn't $x=y$ no matter what?
Sep
22
asked Proving that $\alpha:\mathbb{R}\to\mathbb{R}$ where $\alpha(x)=\frac{x^{3}}{x^{2}+1}$ is bijective
Sep
21
comment Give an example of two functions $\alpha,\beta$ on a set $A$ such that $\alpha\circ\beta=\mathsf{id}_{A}$ but $\beta\circ\alpha\neq\mathsf{id}_{A}$.
I was fixated on finite sets.. this is why I was confused.
Sep
21
asked Give an example of two functions $\alpha,\beta$ on a set $A$ such that $\alpha\circ\beta=\mathsf{id}_{A}$ but $\beta\circ\alpha\neq\mathsf{id}_{A}$.
Sep
21
revised How many reflexive and symmetric relations are in set $A$?
added 35 characters in body
Sep
21
asked How many reflexive and symmetric relations are in set $A$?
Sep
20
comment Determine whether the function $\alpha:A\times B\rightarrow B\times A$ where $\alpha((a,b))=(b,a)$ is injective and/or surjective
@SpamIAm I suppose I missed a step. I intended to say $(b,a)=(d,c)$ as these are the images of $(a,b)$ and $(c,d)$.
Sep
20
revised Determine whether the function $\alpha:A\times B\rightarrow B\times A$ where $\alpha((a,b))=(b,a)$ is injective and/or surjective
edited body
Sep
20
asked Determine whether the function $\alpha:A\times B\rightarrow B\times A$ where $\alpha((a,b))=(b,a)$ is injective and/or surjective