For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.
0
votes
1answer
30 views
How do I prove $\dim{U} = \dim{W}$ when…
$\mathbf{U} = sp\{(a_1 a_2 a_3),(b_1 b_2 b_3),(c_1 c_2 c_3)\}$
$\mathbf{W} = sp\{(a_1 b_1 c_1),(a_2 b_2 c_2),(a_3 b_3 c_3)\}$
$\mathbf{U}$ and $\mathbf{W}$ are subspaces of $\mathbb{R}^3$
0
votes
0answers
37 views
property of an increasing or decreasing function
For $x \in \mathbb{R}$, and $f(x)$ an increasing function, can we prove whether
$$ af(x)\lesseqgtr f(ax) $$
for $a >0$? If we have additional information that $f$ is homogeneous of some degree, ...
2
votes
0answers
52 views
Proving identity involving sum
I'm stuck trying to prove the following identity, which is seemingly correct (from mathematica):
$$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose ...
0
votes
0answers
46 views
Product of Uniform and Gamma Random Variables
Let $X\sim\operatorname{Gamma}(1+\alpha,1)$ and $U\sim \operatorname{U}[0,1]$ be independent, $\alpha < 1$
How do you go about proving that $XU^\frac{1}{\alpha}\sim\operatorname{Gamma}(\alpha,1)$?
...
0
votes
2answers
135 views
Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
22
votes
4answers
571 views
What is a proof?
I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra).
Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
2
votes
3answers
56 views
problem with induction?
I am a bit new to logical induction, so I apologize if this question is a bit basic.
I tried proving this by induction:
$$\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$$
Starting with the base ...
1
vote
2answers
65 views
Law of large numbers?
Given random variables $Z_1,Z_2,Z_3,\ldots$, which are uniformly distributed for $[8,10]$:
If $X_k =\min\{Z_1, Z_2,Z_3,\ldots,Z_k\}$, prove convergence in probability and find the constant. ...
0
votes
2answers
49 views
If $E = \{ x \in \mathbb{R}: \sin(\frac1{x}) = 1\}$ then $l = 0$ is a limit point of E
If $E = \{ x \in \mathbb{R}: \sin\left(\frac{1}{x}\right) = 1\}$, then $l = 0$ is a limit point of $E$.
I have a proof here but I don't quite understand a few points, I hope someone can explain it a ...
2
votes
2answers
52 views
How to prove or statements
How do I prove statements of the following types:
$A \text{ or } B \implies$ C
$A \implies B \text{ or } C$
I don't know how to go about proving statements like this when they have "or". Can ...
-6
votes
2answers
59 views
Help with Theorem III.3.11 in Hungerford's algebra book
I need help to prove part (i) of this theorem which I couldn't prove.
Any help would be appreciated. Thanks in advance.
3
votes
2answers
94 views
$V_\omega$ is countable
Is there an easy way to prove this? I found a book that suggests the injection $h:V_\omega\to\omega$ defined by
$$h(\{x_1,x_2,\dots,x_n\})=2^{h(x_1)}+2^{h(x_2)}+\cdots+2^{h(x_n)},$$
but I hit some ...
6
votes
4answers
61 views
Any commutative associative operation can be extended to a function on nonempty finite sets
This is a fact we use very frequently in general mathematics when we write such notations as $1+2+3+4$: since we know that $+$ is commutative and associative, we can just "drop the parentheses" and ...
0
votes
2answers
50 views
Help in a proof of a result in Hungerford's book
I need help to proof the last part of this corollary:
I didn't understand the part (IV) because the author proves just the canonical projection case and the statement says "every nonzero ...
4
votes
0answers
58 views
Good examples of proofs in mathematics exemplary of creative reasoning [closed]
Just what the title says. I'm not looking for any proofs that require specialized knowledge past the very fundamentals of real analysis. I'm looking for proofs for important results (don't have to be ...
1
vote
0answers
52 views
Struggling with writing logical proofs
I am struggling with the way to write a clear and mathematical proof of logical theorems. Take for example the theorem $\Gamma \models A, \Gamma \subseteq \Delta$ implies $\Delta \models A$. I can ...
2
votes
2answers
52 views
find the largest integer less than a number
I'm trying to figure out this problem for few hours now. please help.
Define $$\text{PCOM}(a,b,c) := \{ax + by + cz : x, y, z ≥ 0\}.$$
Given integers $t > c > b > a > 1,$ I need help ...
4
votes
2answers
76 views
Prove that if a set $E$ is closed iff it's complement $E^{c}$ is open
I was wondering if this proof was right.
$\Leftarrow$ Suppose $E$ is closed. Then choose $x\in E^{c}$, then $x\notin E$, and so $x$ is not a limit point of $E$.
Hence there exists a neighborhood ...
3
votes
1answer
25 views
Preimages of a function: Is the following proposition true or false?
Let $g: ℤ \times ℤ → ℤ \times ℤ$ be defined by $g(m,n) = (2m, m – n)$.
Is the following proposition true or false? Justify your conclusion.
For each $(s, t) ∈ ℤ \times ℤ$, there exists an $(m, n) ∈ ℤ ...
7
votes
1answer
37 views
Show that if some nontrivial linear combination of vectors $\vec{u}$ and $\vec{v}$ is $\vec{0}$, then $\vec{u}$ and $\vec{v}$ are parallel.
I've never been that great at writing proofs, but I'm getting a bit better. I think I have the answer correct, but I don't know if I'm missing anything. My logic seems right but there may be some ...
2
votes
1answer
36 views
Using Archimedean Property to Prove the following
So I've worked through a few of the properties of Archimedas. That is, I understand that for every real number $x$, there exists a natural number $n$ such that $n>x$
And I've also been able to ...
3
votes
1answer
71 views
$AX=C$: An Inconsistent Linear Equation [duplicate]
Question:
Let $A \in M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$
has more than one solution. Prove that there is a column $C \in F^n$ such that
the system of linear ...
3
votes
0answers
64 views
Prime number theorem, proof error
Can someone help me find where I made an error in this attempted proof,
$$M(x)=\sum_{n\leq x}\mu(n)$$
$$\psi(x)=\sum_{n\leq x}\Lambda(n)=\sum_{n\leq x}\ln(n)*\mu(n)=\sum_{n\leq ...
2
votes
3answers
55 views
Set Distributive Property Proof
Prove the distributive property for sets:
$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
I'm not good with proofs but my understanding is that I have to prove 2 things:
(1) $A \cup (B ...
0
votes
1answer
30 views
prove $f^{-1}(B)=A$
I am given $A_1$, $A_2 \subseteq A$ and $B_1$,$B_2 \subseteq B$. and the function $f: A \rightarrow B$
I want to prove that $f^{-1}(B)=A$.
I just assume that here one is talking about ...
1
vote
3answers
101 views
prove: $\dfrac{2^{n+1}+(-1)^n}{3}$
I am asked to prove this notation with induction for $n\in \mathbb{N}$:
real problem is to fill the area with tilings. and for $n\in \mathbb{N}$ there are exactly so many chances to fill the area as ...
2
votes
1answer
24 views
If $pq=1$, then $p=q=1$ for $p,q \in \mathbb {Z}, p,q >0$
If $pq=1$, then $p=q=1$ for $p,q \in \mathbb {Z}$, $p,q >0$
I tried to do this by contradiction
and I get
$(pq=1) \land (p\neq 1 \lor q \neq 1)$
then I have no ideas how to continue with a ...
1
vote
0answers
32 views
Writing Proof Examples
What are examples of magic mountain an circle diagram proofs for 2nd - 5th grade math problems? I want to teach how to prove word problem outcomes of addition and subtraction.
5
votes
3answers
111 views
Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]
The problem is following, prove that:
$$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$
I've tried solving this problem using mathematical induction, but I ...
0
votes
1answer
44 views
Converges in measure then a subsequence converges almost everywhere
Let $(X,\mathcal{M},\mu)$ be a measure space. Let $\{f_n\}_{n\in\mathbb{N}}$ a sequence of measurable functions which converge in measure to a function $f$. Prove that exists a subsequence ...
0
votes
2answers
54 views
If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.
Thanks in advance to anyone who can help me out on this. I'm currently a junior in high school taking and doing well my school's honors pre-calc class, but of all of the math I've ever learned, proofs ...
2
votes
1answer
31 views
Proof the sum of the square of the in and out degree are the same [duplicate]
I know by the handshaking theorem that in a graph, the sum of the in degree and the sum of the out degree will be the same. I observe that in a complete directed graph (as in a complete graph that has ...
1
vote
1answer
51 views
Is this proof on the product of $X$ OK?
Let $X^2$ be star $\sigma$-compact and $F$ be a closed subset in
$X^2$. If $\mathcal{U}$ is an open cover of $F$, then there exists a
$\sigma$-compact subset $A$ of $X$, such that $F \subseteq
...
-2
votes
1answer
51 views
Given the following recurrence relation, prove using mathematical induction
How can we prove this using mathematical induction?
$m_1 = 0$
$m_k = m_{\lfloor (k/2) \rfloor} + m_{\lceil (k/2) \rceil} + k-1$ for all integers $k \geq 1$
Prove using mathematical induction that ...
0
votes
6answers
110 views
Finding the number of subsets of S
How can we find the number of subsets of $S=\{1,2,3,...,10\}$ that contain neither 5 nor 6?
Thanks!
0
votes
2answers
49 views
Use the binomial theorem to expand
How can we expand this using the binomial theorem?
$(x^2 + 1/x)^7$
0
votes
1answer
30 views
Subsequence proof
"Let $\{a_n\}$ be a sequence. Let $\{b_{n_k}\}$ and $\{c_{n_k}\}$ be subsequences of $\{a_n\}$. Prove directly that if $b_{n_k} \to b$ and $c_{n_k} \to c$ with $b \neq c$, then $\{a_n\}$ does not ...
27
votes
5answers
3k views
Are all prime numbers finite?
If we answer false, then there must be an infinite prime number. But infinity is not a number and we have a contradiction. If we answer true, then there must be a greatest prime number. But Euclid ...
2
votes
3answers
51 views
Proofs and Number theory
I am needing help proving the following:
For any integer $n$, $n^2$ + 5 is not divisible by $4$
I am aware that an integer $x$ is divisible by integer $y$ if there exists integer $k$ such that ...
2
votes
1answer
37 views
Can one use Pick's theorm to prove that area size 5 covers at least 6 grid points?
According to Pick's Theorem, the size of an area $A$ can be calculated by the sum of
the interior lattice points located in the polygon $i$ and the number of lattice points on the boundary placed on ...
1
vote
1answer
30 views
Let $X$ has countable extent. Does $X^2$ have countable extent?
Definition 1: A space $X$ has countable extent if every uncountable subset of $X$ has a limit point in $X$.
I'm struggling with this question:
Question 2: Let $X$ has countable extent. Does ...
5
votes
5answers
121 views
Prove that $\log X < X$ for all $X > 0$
I'm working through Data Structures and Algorithm Analysis in C++, 2nd Ed, and problem 1.7 asks us to prove that $\log X < X$ for all $X > 0$.
However, unless I'm missing something, this can't ...
1
vote
2answers
42 views
Let H and K be subgroups of the finite group G and supposes $|H|^{2}>|G|$ and $|K|^{2}>|G|$. Prove $H \cap K$ has at least two elements
So I supposed $|H \cap K|>1$
$\Rightarrow |HK||H \cap K|> |HK|$
Which eventually implied that
$\Rightarrow |H \cap K|>|G|$
Thus since G is a group, and H and K are subgroups then the ...
1
vote
2answers
29 views
In order for a proposition to be completely stated, how much detail needs to go into describing it?
During my study of mathematics so far, I had come to realize that how a proposition is stated can be just as important as the proof itself.
For example, when I was working on various propositions on ...
7
votes
2answers
180 views
maximum number of edges to be removed to possess a property
I am working on a problem. We know that on squaring a cycle, degree of every vertex is 4. For squares of cycles, we know if we delete any arbitrary edge then still eccentricity is same for all ...
0
votes
2answers
50 views
Help with writing proofs
Prove that for any sets A, B and C if A is a subset of B, then A – C is a subset of B – C.
1
vote
2answers
36 views
How to do a combinatorial proof
I have a question which asked for a combinatorial proof. I have no clue how to do do a combinatorial proof.
The question is
prove that the total number of subsets in $\{x_1, x_2, x_3, ... ,x_n\}$ is ...
2
votes
3answers
77 views
Proof of $\;\text{Asymmetric}(\sqsubset)\rightarrow \text{Antireflexive}(\sqsubset)$
The relation $\;\sqsubset\;\subseteq S\times S$ is asymmetric if
$$\forall a,b\in S:(a,b)\in\sqsubset\rightarrow (b,a)\notin\sqsubset$$
and it is antireflexive if
$$\forall a\in ...
4
votes
4answers
113 views
If $S_n = 1+ 2 +3 + \cdots + n$, then prove that the last digit of $S_n$ is not 2,4 7,9.
If $S_n = 1 + 2 + 3 + \cdots + n,$ then prove that the last digit of $S_n$ cannot be 2, 4, 7, or 9 for any whole number n.
What I have done:
*I have determined that it is supposed to be done with ...
2
votes
1answer
54 views
how to show associativity of multiplication for not just 3 operands but for n operands
ie Id like to show
a(bc)=(ab)c
but for any n operands
eg
abcdefg=gfdcabe etc
I can see this is very intuitive that this should be true for all n operands, but as a logical exercise I would like to ...
