Intereting Posts

Are there any elegant methods to classify of the Gaussian primes?
Largest value in the functional $\int_0^\infty e^{-rt}( x^2+2x+\dot x^2)dt$?
Is the image of a nowhere dense closed subset of $$ under a differentiable map still nowhere dense?
Can $e_n$ always be written as a linear combination of $n$-th powers of linear polynomials?
Indiscrete rational extension for $\mathbb R$ (examp 66 in “Counterexamples in topology”)
How prove this inequality $\sum\limits_{cyc}\frac{a^2}{b(a^2-ab+b^2)}\ge\frac{9}{a+b+c}$
Prove that if $A$ is compact and $B$ is closed and $A\cap B = \emptyset$ then $\text{dist}(A,B) > 0$
Compact group actions and automatic properness
What is meant by limit of sets?
$\frac{\partial T}{\partial t} = \alpha \nabla ^2r$ for spherically symmetric problems
Proving that $\gcd(2^m – 1, 2^n – 1) = 2^{\gcd(m,n )} – 1$
Smooth functions with compact support are dense in $L^1$
Proof of another Hatcher exercise: homotopy equivalence induces bijection
Proving that the estimate of a mean is a least squares estimator?
Showing non-cyclic group with $p^2$ elements is Abelian

Possible Duplicate:

Injective and Surjective Functions

If $g(f(x))$ is one-to-one (injective) show $f(x)$ is also one-to-one given that $f$ is a function from $A$ to $B$ and $g$ a function from $B$ to $C$.

I’ve just started my Discrete math course and I’d like some help on this. I’m pretty sure we’re supposed to use set theory laws to prove this.

- Is there a continuous function $f(x)$ such that the inverse function is $1/f(x)$?
- If $f\circ f\circ f=id$, then $f=id$
- Sufficient / necessary conditions for $f \circ g$ being injective, surjective or bijective
- Existence and uniqueness of a function generalizing a finite sum of powers of logarithms
- Codomain of a function
- Injective map from real projective plane to $\Bbb{R}^4$

So far I know the three conditions that satisfy an injective function (sorry, having difficulties typing all this TeX markup so I’ll skip that).

Any help?

- Roots of a polynomial whose coefficients are ratios of binomial coefficients
- Are $L^\infty$ bounded functions closed in $L^2$?
- Drawing by lifting pencil from paper can still beget continuous function.
- When do two functions become equal?
- If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$.
- What is the limit distance to the base function if offset curve is a function too?
- Prove that $C = f^{-1}(f(C)) \iff f$ is injective and $f(f^{-1}(D)) = D \iff f$ is surjective
- second derivative of the inverse function
- Describing a Wave
- Axiom of Choice and Right Inverse

Hint: suppose that $f$ is not one to one. Then there are $x \ne y$ such that $f(x)=f(y)$. Can you conculude that $g\circ f$ is not one to one?

Or the other way around:

Take $x \ne y$. Since $g \circ f$ is injective we have $g(f(x))\ne g(f(y))$, so we must have $f(x)\ne f(y)$.

Hence $g \circ f$ is injective then by definition it means that for all $x, y$ in the domain of $f$ (being the domain of $g \circ f$) we get

$$[(g \circ f) (x) = (g \circ f)(y)] \Rightarrow x = y.$$

Assume now that $f(x) = f(y)$. Then of course $(g \circ f)(x) = g(f(x)) = g(f(y)) = (g \circ f)(y)$. By previous implication we obtain $x = y$ and hence $f$ is injective.

- How to define a well-order on $\mathbb R$?
- Would this proof strategy work for proving the lonely runner conjecture?
- proving identity for statistical distance
- Rigorous definition of “Differential”
- How to get used to commutative diagrams? (the case of products).
- How to create alternating series with happening every two terms
- Proving if $F^{-1} $ is function $\Rightarrow F^{-1}$ is $1-1$?
- Two complicated limits: $\lim_{x\to 0}\frac{e^{ax}-e^{bx}}{\sin(cx)}$ and $\lim_{x\to 0} x(a^{\frac1x}-1)$
- $f_n(x)$ convergence in measure implies $\frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}$ convergence almost everywhere
- Is $\lim\limits_{x\to x_0}f'(x)=f'(x_0)$?
- Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?
- Probability distribution of sign changes in Brownian motion
- separation properties in Hausdorff, compact spaces
- Differentiability-Related Condition that Implies Continuity
- Show that $k/(xy-1)$ is not isomorphic to a polynomial ring in one variable.