The proof-strategy tag has no wiki summary.
-3
votes
2answers
68 views
If $P(n)$ is true for $n = 1$, if $P(n)\implies P(2n)$, and if $P(n) \implies P(n - 1)$, then $P(n)$ is true for all $n$. [closed]
Assume that the statement P(n) is true for n = 1 and that P(n) → P(2n) for all n ∈ ℕ. Furthermore assume that if n > 0 then P(n) → P(n - 1). Prove (∀n ∈ N)P(n). The following two exercises deal with ...
2
votes
2answers
36 views
Tests/ invariants for module isomorphisms
It two modules are indeed isomorphic, then it is often not too difficult to find an isomorphism since most of the time it is just the natural map. However, it takes some time for me to prove that two ...
5
votes
5answers
114 views
Proving $n+3 \mid 3n^3-11n+48$
I'm really stuck while I'm trying to prove this statement:
$\forall n \in \mathbb{N},\quad (n+3) \mid (3n^3-11n+48)$.
I couldn't even how to start.
0
votes
2answers
67 views
Mathematical Proof Question?
How would prove that this is true for all $k \geq 6$:
$$\left(\frac{1+\sqrt{5}}{2}\right)^{k+1} - \left(\frac{1-\sqrt{5}}{2}\right)^{k+1} \geq (1.5)^{k} \times \sqrt{5} $$
1
vote
2answers
121 views
Proof Help: Membership Table
I am new to proofs with membership tables and this is the last question I am posting.
I am trying to teach myself discrete math and am stuck on this:
Let $ A, B$ and $C$ be sets in the universal ...
3
votes
5answers
185 views
Proof: How to prove $n$ is odd if $n^2 + 3$ is even
New to the whole proof thing.
Trying to figure out that, for all integers $n$, if $n^2 + 3$ is even, then $n$ is odd.
Thank you for the help.
2
votes
3answers
106 views
Show that if $n$ and $k$ are integers with $1 ≤ k ≤ n$, then ${n\choose k} \le (n^k)/ 2^{k−1}$
I've looked everywhere but I've been unable to come up with a way to show that if $n$ and $k$ are both integers such that $1 ≤ k ≤ n$, then:
$${n \choose k} \le \frac{n^k}{2^{k−1}}$$
Thank you!
1
vote
2answers
21 views
Let $a\in\mathbb R^n$ be a fixed point. How to prove $B(a, 1/2)\cap B(g+a, 1/2)=\emptyset$ where $g\in \mathbb Z^n-\{0\}$?
Let $a\in\mathbb R^n$ be a fixed point. How to prove $B(a, 1/2)\cap B(g+a, 1/2)=\emptyset$ for some $g\in \mathbb Z^n-\{0\}$?
It sounds a silly question, and obvious one, but it is a fact I need for ...
2
votes
1answer
25 views
Proof for a creating a partition of a countable set using chains in partial orders.
Definition: A partition of a set $A$ is a set of nonempty subsets of $A$ called the
blocks of the partition, such that
every element of $A$ is in some block, and
if $B$ and $B'$ are different ...
2
votes
2answers
119 views
Exponential function formula proof
How does one arrive at $e^4$ from
$$\sum_{x=0}^{\infty}\frac{ 4^x}{x!}$$
3
votes
2answers
212 views
Prove that such an inverse is unique
Given $z$ is a non zero complex number, we define a new complex number $z^{-1}$ , called $z$ inverse to have the property that $z\cdot z^{-1} = 1$
$z^{-1}$ is also often written as $1/z$
0
votes
0answers
59 views
A procedure for Topological sort, proof for its correctness.
Definition: A preserved invariant of a state machine is a predicate, $P$, on
states, such that whenever $P(q)$ is true of a state, $q$, and $q \rightarrow r$ for some state, $r$,
then $P(r)$ holds.
...
0
votes
2answers
35 views
How do you explain $f(x_4|x_3)f(x_3|x_2)f(x_2|x_1)f(x_1) = f(x_4,x_3,x_2,x_1)$?
Let $x_1=x(n_1)$, $x_2=x(n_2)$, $x_3=x(n_3)$ and $x_4=x(n_4)$ be random Markov processes $(n_1 < n_2 < n_3 < n_4)$.
I don't understand the identity given below on their probability density ...
3
votes
1answer
53 views
line of mathematicians guess their own hat color out of c colors
There is a common problem in which a long line of N mathematicians are each given a hat that is either red or blue. They cannot see their own hat but can see all in front of time and can hear any ...
3
votes
1answer
46 views
An ordering different from the Gray order (digits change by 1 at each step)
Given $A=\lbrace x_n,\ldots x_1\rbrace$. How would I construct an ordering on the subsets of $A$ such that the immediate successor of a subset is obtained by either adding or deleting one element, and ...
2
votes
3answers
66 views
Is the sum or product of idempotent matrices idempotent?
If you have two idempotent matrices $A$ and $B$, is $A+B$ an idempotent matrix?
Also, is $AB$ an idempotent Matrix?
If both are true, Can I see the proof? I am completley lost in how to prove both ...
1
vote
3answers
58 views
Generating function with quadratic coefficients.
$h_k=2k^2+2k+1$. I need the generating function $$G(x)=h_0+h_1x+\dots+h_kx^k+\dots$$ I do not have to simplify this, yet I'd really like to know how Wolfram computed this sum as ...
0
votes
0answers
44 views
Prove correctness of the algorithim
Consider the following algorithm min which takes lists x,y as parameters and returns the zth smallest element in union of x and y.
Pre conditions: X and Y are sorted lists of ints in increasing order ...
0
votes
1answer
43 views
Subring Proof Question
The question reads: "Let $T$ be the ring of all continuous functions from $\Bbb{R}$ to $\Bbb{R}$. Let $S = \{f \in T \mid f (2) = 0\}$. Either prove or disprove that $S$ is a subring of T".
(Note: E ...
1
vote
1answer
27 views
Correctness and help with Union and intersection proof of Open Sets
I need to prove the following:
Let $A$ and $B$ be subsets of a metric space $(X, d)$ show that
$A^o \cup B^o \subset (A \cup B)^o$.
$(A \cap B)^o = A^o \cap B^o$
Here is my attempt:
For 1)
Let ...
9
votes
4answers
289 views
Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$
If $a,b,c$ are non-negative numbers and $a+b+c=3$, prove that:
$$\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$$
Here's what I've tried:
Using Cauchy-Schawrz I proved that:
$$(3a + ...
1
vote
0answers
36 views
Find the number of permutations in these words
Finding the number of permutations in these three words, am I doing this correctly?
a) CORRECT = $\frac{7!}{2!\cdot2!} = 1260$
b) COEFFICIENT = $\frac{11!}{2!\cdot2!\cdot2!\cdot2!} = 2494800$
c) ...
1
vote
1answer
18 views
How many permutations of this set can be made?
How many permutations of the set of seven letters (A,B,C,D,E,F,G) have the two vowels before the five consonants?
I'm wondering here if we use the set of 7! - 2! since they can only occupy the first ...
1
vote
3answers
118 views
Showing that $3n<n!$ whenever $n$ is an integer with $n \geq 7$
How can we show that:
$$3n< n!$$
whenever $n$ is an integer such that $n \geq 7$ ?
I was thinking that we can prove this by showing that such case is true with any integer above 7, but ...
1
vote
3answers
86 views
How many different permutations of this set don't have vowels on the ends?
If we have the set of seven letters: (A,B,C,D,E,F,G) then how many permutations of these seven letters do not have vowels on the ends (that is, both the first and last letters are consonants)? I was ...
6
votes
4answers
163 views
Proof: if the graphs of $y=f(x)$ and $y=f^{-1}(x)$ intersect, they do so on the line $y=x$
This came out of a textbook problem, and as Lubin pointed out below, it's not actually true as originally stated. I'm guessing it should be restated as: If the graphs of $y=f(x)$ and $y=f^{-1}(x)$ ...
1
vote
1answer
55 views
Formal deduction in first order logic
How do you show that a deduction exist in the Hilbert Proof System, as used in Enderton's Mathematical Introduction To Logic.
L is a FOL (First Order Language).
L contains R, where R is a single ...
0
votes
1answer
38 views
Math Analysis - Two Bounded Functions and partitions proof
Let $f,g:[a,b]\rightarrow \mathbb R$ be bounded functions and $g$ is increasing. Show that for every partitions $P$ and $Q$ of $[a,b]$ with $P\le Q$ we have $s(f,P,g) \le s(f,Q,g) \le S(f,Q,g)\le ...
1
vote
1answer
30 views
Pointers about the concept of 'division extensionality'?
When working a bit on another question (If $a \equiv b\pmod m$, then $\gcd(a, m) = \gcd(b, m)$), I discovered the following, which seems to be valid:
$$
a = b \;\;\equiv\;\; \langle \forall d :: d ...
2
votes
1answer
114 views
Convergence of a power series function
Consider the following differential equation:
$$w''(x)+p(x)w'(x)+q(x)w(x)=r(x)$$
with the initial condition of $w(0)=w_0,\ w'(0)=w_1$, and
$$w_{n+2}=\frac{r_{n+2}-(n+1)p_0w_{n+1}-\sum_{k=0}^n w_k ...
2
votes
1answer
95 views
Providing a sketch for a proof before proceeding through the actual proof. [closed]
Question is pretty straightforward. My mathematics is sloppy, and I recognize my inaptitude in that my proofs are more or less too intuitive. My diagnosis dictates
the fact that I attack a problem ...
1
vote
0answers
50 views
Examining every mathematical result in purely formal, ZFC language.
My main interest is physics. However, being self-taught in mathematics for the most part, my proofs tend to be more intuitive than it is acceptable. Yet, I recognize my inaptitude in rigor, and I ...
1
vote
2answers
75 views
Convergent sequences and proof
Prove that $\dfrac{1+n}{n^2}$ converges as $n \to \infty$
How do I go about constructing this proof? Can I use the definition that $\operatorname{abs}(a_n - L < \epsilon)$?
2
votes
3answers
91 views
Prove$\overline{(A \cap B \cap C)} = \overline{A} \cup \overline{B} \cup \overline{C}$ By Subsets
This problem I am trying to solve is one I alluded to in this thread: Proving By Subsets
I am having difficulty with proof by subsets, so I am aware that I am missing steps; I would certainly ...
-2
votes
1answer
91 views
Proving By Subsets [closed]
I am currently trying to learn about conducting proofs by using subsets. In my textbook, there is an example on this very matter; however, the seem to do something that is in contradiction with what ...
2
votes
2answers
76 views
Is this proof using the pumping lemma correct?
I have this proof and it goes like this:
We have a language $L = \{\text{w element of } \{0,1\}^* \mid w = (00)^n1^m \text{ for } n > m \}$.
Then, the following proof is given:
There is a $p$ ...
0
votes
1answer
101 views
Induction proof on covering a checkerboard with dominoes - don't think my proof is right.
So I'm trying to solve this problem and I think I'm on the write track, but my proof relies on a domino being divisible by 2, which I don't think is correct.
The problem:
Prove that a $2^n \times ...
8
votes
2answers
174 views
Prove that $\mathcal{P}(A)⊆ \mathcal{P}(B)$ if and only if $A⊆B$. [duplicate]
Here is my proof, I would appreciate it if someone could critique it for me:
To prove this statement true, we must proof that the two conditional statements ("If $\mathcal{P}(A)⊆ \mathcal{P}(B)$, ...
5
votes
4answers
124 views
Proving that $\lim_{h\to 0 } \frac{b^{h}-1}{h} = \ln{b}$
Is there a formal proof of this fact without using L'Hôpital's rule? I was thinking about using a proof
of this fact:
$$
\left.\frac{d(e^{x})}{dx}\right|_{x=x_0} = e^{x_0}\lim_{h\to 0} ...
0
votes
0answers
22 views
Equivalent metrics and inclusion of balls [duplicate]
I need some help with the following proof. I am stuck. My general idea is that if $d_1$ and $d_2$ are equivalent metrics then the balls converge to the same point? However, my understanding of metric ...
3
votes
2answers
43 views
If $x\lt y $ for arbitrary real x and y there exists a real r $r$ such that $x \lt r \lt y$ and hence infinitely many.
If $x\lt y $ for arbitrary real $x$ and $y$ there exists a real r $r$ such that $x \lt r \lt y$
Prove that there is at least one r satisfying this inequality, and hence infinitly many.
I was ...
1
vote
3answers
124 views
Proving That The Product Of Two Different Odd Integers Is Odd
Okay, here is how I begin my proof:
Let $q$ and $r$ be odd integers, then $q = 2k+1$ and $r = 2m+1$, where $k,m \in Z$.
$q \times r = (2k+1)(2m+1) \implies q \times r = 4mk + 2k + 2m + 1 \implies q ...
4
votes
2answers
176 views
Proof by contradiction: $ \emptyset \subseteq A$
I have to proof by contradiction that: let $ A $ a set and $ \emptyset $ the empty set, then $ \emptyset \subseteq A$; if $ \emptyset \nsubseteq A$ then $\exists x \in \emptyset ( x \notin A ) $ ...
0
votes
1answer
40 views
Showing the following language is not contex free
I need to show the following language is not context free via the Pumping Lemma.
$$L = \{0^n\#0^{2n}\#0^{3n}\mid n \ge 0 \}$$
I was wondering if someone can help explain how to begin such a proof. ...
2
votes
3answers
74 views
proof of combinatoric/using pascals theorem
prove that, for even values of $n$, $$\sum_{i=0}^{n/2}\binom{n}{2i}= 2^{n-1}\;.$$
I tried using pascals theorem to help prove this with no success
2
votes
1answer
28 views
How to show any space is dually scattered?
As the title explains, how to show any space is dually scattered?
A topological space $X$ is dually scattered if for any neighbourhood assignment $\{ O_x : x \in X \}$ there is a scattered ...
0
votes
1answer
59 views
Einstein Summation Proof, Does this count as an expansion?
Prove curl grad $\phi = 0$ for all scalar fields. $\phi$ (i.e. $\nabla \times \nabla \phi \ 0$ using Einstein summation notation only, no term by term expansion.
Well, first step is to set up in ...
4
votes
0answers
51 views
Arbitrary ratio sequences on a partition of $\mathbb{R}$ (Partition regularity of fixed ratio sequences)
Background:
This question arose purely recreationally and doesn't really fit into any context that I know of.
Let $A \sqcup B = \mathbb{R}$ be a partition of the ...
2
votes
1answer
46 views
Magnitude of a vector, cubed, in Einstein Summation Notation
Evaluate $\nabla \cdot (r^3v)$. The answer will be in terms of r.
Where v represents the position vector and r represents the scalar magnitude of the position vector.
I started by writing this in ...
2
votes
1answer
100 views
Can I use Schwartz's Lemma to prove that $f(0)=0$ and $\operatorname{Re}f(z)\rightarrow 0$ implies $f(z)=0$ for all $z\in\mathbb{C}$?
Problem. Suppose that $f(x)$ is an entire function satisfying $f(0)=0$ and $\operatorname{Re}f(z)\rightarrow 0$ as $|z|\rightarrow \infty$. Show that $f(z)=0$ for all $z\in \mathbb{C}$.
The ...
