I want to prove $\alpha \rightarrow \beta, (\alpha \rightarrow \lambda) \rightarrow \beta $ proves $\beta$ in HPC  I use MP and the axioms listed here:   I guess that the end of the proof should look something like this:  $(\alpha\rightarrow(\beta\rightarrow\lambda)) \rightarrow ((\alpha \rightarrow \beta) \rightarrow (\alpha \rightarrow\lambda))$  $(\alpha \rightarrow \beta) \rightarrow (\alpha \rightarrow\lambda)$   $\alpha \rightarrow \lambda$  $\beta$  but I can't find out how I can get to $(\alpha\rightarrow(\beta\rightarrow\lambda))$ given my premises  would be happy for some help
124257
I'd really like your help proving:  $(A\rightarrow B),(A\rightarrow C)\rightarrow B,  \mapsto_{HPC} B $  Where $HPC$ is the Hilbert's system proof which contains the following relevant axioms:   $A\rightarrow(B \rightarrow A)$ $(A\rightarrow(B\rightarrow C)) \to ((A\rightarrow B)\rightarrow(A\rightarrow C))$ $(A\rightarrow B)\rightarrow ((A\rightarrow\bar{B})\rightarrow \bar{A})$ $\bar{\bar{A}} \rightarrow A$   In addition tried to use these following lemmas: $\bar{A} \rightarrow (A \rightarrow C) $ and $(A\rightarrow B)\rightarrow (\bar{B} \rightarrow \bar{A})$.  Any suggestions?
80261
If $M$ is a closed manifold, it's well known that $\Delta\phi=\alpha$, where $\alpha$ is a $p$-form, has a solution if and only if $\alpha$ is orthogonal to the space of harmonic $p$-forms. This is proved on page 224 of Warner, but the proof depends heavily on his approach to Hodge theory. There are other methods that avoid his main technical lemma 6.6. So, can one prove this fact using only the "standard" facts of the theory?  For functions it is easy, for one has $$\alpha=\delta \psi+h.$$ But $(\alpha,h)=0$, so $\alpha=\delta\psi$. Then decompose $\psi$ to get $\alpha=\delta d\zeta$ for some function $\zeta$. But on functions, $\Delta=\delta d$. But for $p$-forms in general one has $$\alpha=d\xi+\delta \psi+h,$$ since $\Omega^{p-1}$ is nontrivial. Using the decompositions again, one has $$\alpha=d\delta \rho+\delta d\sigma,$$ where $\rho,\sigma$ are $p$-forms. I don't see how $\rho$ and $\sigma$ are related, but it looks close.
96623
I think it's true, because I can't see counterexamples.  Here's a proof that I am not sure of:  Let $p_1,p_2,\ldots, p_n$ be the prime factors of $a$ or $b$  \begin{eqnarray} a&amp;=&amp; p_1^{\alpha_1}\cdots p_n^{\alpha_n} \\ b&amp;=&amp; p_1^{\beta_1}\cdots p_n^{\beta_n} \\ a^3\mid b^2 &amp;&amp;\Rightarrow \frac{b^2}{a^3} \in \mathbb{Z} \\  &amp;&amp;\Rightarrow 2 \beta_i - 3\alpha_i \geq 0 \\ &amp;&amp;\Rightarrow 2 \beta_i \geq 3\alpha_i \text{ and } \beta_i \geq 0 \\ &amp;&amp;\Rightarrow 3 \beta_i \geq 3\alpha_i \\ &amp;&amp;\Rightarrow \beta_i \geq \alpha_i \\ &amp;&amp;\Rightarrow \beta_i - \alpha_i \geq 0 \\ &amp;&amp;\Rightarrow \frac{b}{a} \in \mathbb{Z}\\ &amp;&amp;\Rightarrow a\mid b \end{eqnarray} Is this proof correct ?
120977
I am having trouble with the following problem:   $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent.   Work: I am told that Euclid's Lemma will be useful to proving injection and also that for any positive integer $n$, $n$ is a product of prime numbers.   For injection, I set $f(a,b)=f(x,y)\rightarrow 2^{a-1}(2b-1)=2^{x-1}(2y-1)$.  Then I distributed across to get $2^ab-2^{a-1}=2^xy-2^{x-1} \rightarrow 2^ab-2^xy=2^{a-1}-2^{x-1} \rightarrow 2(2^{a-1}b-2^{x-1}y)=2^{a-1}-2^{x-1}$.   From here, I am unsure how to apply Euclid's Lemma to arrive at the injection result.  I am even more unsure how to prove surjection.
113746
Show that for $u,v \in \mathbb{C}$ with $|u|&lt;1, |v|&lt;1$, and $\bar{u}v\neq u\bar{v}$, we always have $$\left|\left(1+|u|^2\right)v-\left(1+|v|^2\right)u\right|&gt;\left|u\bar{v}-\bar{u}v\right|.$$
7756
Problem. Let $(\mathscr{L},\vdash)$ be a logic and $\Gamma\subseteq\mathscr{L}$ be a set of formulas such that for some $\varphi\in\mathscr{L}$ we have $$\Gamma\vdash\varphi\to\neg\varphi$$ $$\Gamma\vdash\neg\varphi\to\varphi$$Then show that, $$\Gamma\vdash(\varphi\to\neg\varphi)\to\varphi$$using only the following, $\color{crimson}{\text{Axiom 1.}}\  P\to (Q\to P)$ $\color{crimson}{\text{Axiom 2.}}\  (S\to (P\to Q))\to((S\to P)\to (S\to Q))$ $\color{crimson}{\text{Axiom 3.}}\  (\neg Q\to\neg P)\to(P\to Q)$ $\color{crimson}{\text{Rule of Inference.}}$ . $\color{crimson}{\text{Theorem.}}$ .  I have tried for some hours to find a proof of the claim but couldn't succeed. Can anyone help?
32552
Let $\kappa , \lambda$ be cardinals with $\omega \leq \lambda \leq \kappa.$   Prove $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$.  i.e could anyone advise me on how to construct a bijection between $\{f \mid \text{$f$ is a function},~{\rm dom}(f) = \lambda,~ {\rm ran}(f) \subseteq \kappa\}$ and $\{X: X \subseteq \kappa, |X|=\lambda\}$?  Thank you.
12307
I've tried deriving a proof for this using the handshaking lemma, but that was unsuccessful. Could someone please explain to me how they would do this proof?
72014
I have been trying to prove that the propositional formula $ \big( \alpha \rightarrow \lnot \beta \big) \rightarrow \big((\alpha \rightarrow \beta) \rightarrow \lnot \alpha \big)$ is a theorem in Hilbert's system, i.e., to prove $\vdash \big( \alpha \rightarrow \lnot \beta \big) \rightarrow \big((\alpha \rightarrow \beta) \rightarrow \lnot \alpha \big)$using the following three axioms and Modus Ponens.   $\alpha \rightarrow \big(\beta \rightarrow\alpha \big)$ $ \big({\alpha \rightarrow \big(\beta \rightarrow\gamma \big)}\big) \rightarrow  \big({\big(\alpha \rightarrow\beta \big) \rightarrow \big(\alpha \rightarrow\gamma \big)} \big)$ $\big( \lnot \beta \rightarrow \lnot \alpha \big) \rightarrow \big( (\lnot \beta \rightarrow\alpha) \rightarrow \beta \big)$    Deduction theorem can be used as well. I am getting into long winded derivations which don't lead anywhere. Hints will be highly appreciated.
81154
Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$.
119567
How do I prove $$S\rightarrow \exists xP(x) \vdash \exists x(S\rightarrow P(x))$$ using natural deduction? Just an alignment of which axioms or rules that one could use would be much appreciated.
50384
If $K$ and $k$ are fields, $K\supset k$ is a field extension and $\alpha \in K$ is algebraic over $k$, then we denote by $k[\alpha]$ the set of elements of $K$ which can be obtained as polynomial expressions of $\alpha$.   $$k[\alpha] = \left\{P(\alpha): \quad P\in k[X]  \right\} $$  Also, we denote by $k(\alpha)$ the smallest subfield of $K$ containing both $k$ and $\{\alpha\}$. This is easily seen to be equal to the set of fractions of polynomial expressions in $\alpha$ (just thinking about the way the "smallest" subfield must be generated):  $$k(\alpha) = \left\{\dfrac{P(\alpha)}{Q(\alpha)}: \quad P,Q\in k[X] \text{  and } Q(\alpha)\neq 0 \right\} $$  If $M$ is the minimal polynomial of $\alpha$, it is easily shown that $k[\alpha]\simeq k[X]/\langle M\rangle$ is a field  and therefore a subfield of $k(\alpha)$ containing both $k$ and $\{\alpha\}$. Therefore $k(\alpha)=k[\alpha]$, because $k(\alpha)$ is the smallest.   It follows that all quotients $\dfrac{P(\alpha)}{Q(\alpha)}$ are equal to $H(\alpha)$ for some polynomial $H\in k[X]$.   What I want to see is a more direct proof of this fact. Given $P$ and $Q$, how do you produce this $H$?
63370
I couldn’t find any proof on the internet. I would appreciate an elementary proof , but any help would be appreciated!  Thank you.
44541
I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.
79051
Let $\alpha$ be a root of $x^3 + x + 1 \in \mathbb{Z}_2[x]$ and $\beta$ a root of $x^3 + x^2 + 1 \in \mathbb{Z}_2[x]$. Then we know that $$\mathbb{Z}_2(\alpha) \simeq  \frac{\mathbb{Z}_2[x]}{(x^3 + x + 1)} \simeq \mathbb{F}_8 \simeq \frac{\mathbb{Z}_2[x]}{(x^3 + x^2 + 1)} \simeq \mathbb{Z}_2(\beta). $$   I need to find an explicit isomorphism $\mathbb{Z}_2(\alpha) \to \mathbb{Z}_2(\beta). $   I was thinking of finding a basis for $\mathbb{Z}_2(\alpha)$ and $\mathbb{Z}_2(\beta)$ over $\mathbb{Z}_2$.   I let $\left\{1, \alpha, \alpha^2\right\}$ and $\left\{1, \beta, \beta^2\right\}$ be these two bases. Now suppose I have a field morphism $$ \phi: \mathbb{Z}_2(\alpha) \to \mathbb{Z}_2(\beta) $$ which maps $1$ to $1$. how can I show that the image of $\alpha$, i.e. $\phi(\alpha)$, completely determines this map?
20631
Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.  A hint is given that for $0 \le \alpha \le \beta$ there exist a number $\xi \in (\beta, \alpha + \beta)$ s.t. $$(\alpha + \beta)^p - \beta^p = p \xi^{p-1}\alpha \le \alpha^{p}$$.  I see that by proving the above hint the result holds for positive numbers.  In order to prove the above hint I've considered the Intermediate value theorem, however I've not yet succeeded.
35953
I'm looking for proof that  Let $a_{0},a_{1}&gt;0$,and such $$a_{n+2}=\dfrac{2}{a_{n+1}+a_{n}}$$  show that: there exist $\lambda,c&gt;0$ such $$|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$$  I tried using induction, but i'm not sure how it would work ,Thanks in advance.