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.

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0
votes
2answers
124 views

Probability of triangles in $G(n, 1/2)$ [closed]

I am trying to solve the following problem: Let $G = G(n, \frac{1}{2})$ be a random graph on $n$ vertices, i.e., for each pair of vertices $i, j$, we add the edge $(i, j)$ independently with ...
0
votes
2answers
74 views

Evaluating probability as n tends toward infinity [closed]

I feel like this should be a relatively easy proof, but am stumped on how to go about solving it... Any help would be appreciated! Let $S = ${$1,2,\dots,n$}, and let $A,B$ be two random subsets of ...
0
votes
0answers
74 views

Proof about edge crossings

Consider an arbitrary drawing (with possibly intersecting edges) of the complete graph $K_n$ on the plane ($n \geq 5$). I am looking to prove that at least $\frac{1}{5} {n \choose 4}$ pairs of edges ...
0
votes
1answer
24 views

Suppose that $f: X \rightarrow Y$ is a function and $ B \subseteq Y$ is countable then $ f^{-1} $(B) is countable.

Suppose that $f: X \rightarrow Y$ is a function and $ B \subseteq Y$ is countable then $ f^{-1} $(B) is countable. My definition of countable is the following: A non-empty set $X$ is said to be ...
-1
votes
1answer
27 views

If $n \in \mathbb{N}$ , not divisible by 3 show there $\exists t \implies 3^t < n < 3^{t+1}$

If $n \in \mathbb{N}$ , not divisible by 3 show there $\exists t \implies 3^t < n < 3^{t+1}$ By the division algorithm: $$n = 3a + r \implies 0 < r < 3$$ For some $a$ But I cannot do ...
1
vote
1answer
41 views

Elementary proof of Hölder´s inequality (by convexity)

I am trying to prove the Hölder inequality but following these steps 1) Let $f:(a,b)\to\mathbb R$ be double differentiable then $f$ is convex if and only if $\;\;f''(x)>0$ for all $x\in (a,b)$ 2) ...
0
votes
1answer
39 views

Prove that the second derivative is positive iff the function is convex.

Well, I want to prove the following: Let $f:(a,b)\to\mathbb R$ be double differentiable then $f$ is convex iff $\;\;f''(x)>0$ for all $x\in (a,b)$. Then I tried te following: $\Rightarrow]$ Lets ...
0
votes
0answers
15 views

Would someone check this proof about Lipschitz functions?

Let $\bar f(t, \bar y): \mathbb R^{n+1} \supset D\mapsto \mathbb R^n$, we say that it is lipschitz on $D$ with respect to $\bar y$, uniformly in $t$, if exists $L$ such that $$|| \bar f(t, \bar y) - ...
0
votes
3answers
27 views

Suppose that $f: X \rightarrow Y$ is a one to one function and $A \subseteq X$ then $Y - f(A) \subseteq f(X-A)$. Am I on the right track?

Suppose that $f: X \rightarrow Y$ is a one to one function and $A \subseteq X.$ $A, X, Y$, are all sets. I am trying to decipher if the following statements are true or false. If true I will need ...
2
votes
2answers
56 views

How to use Induction with Sequences?

I have posted this similar question here, but with no hopes. I would just like to know: Most of the solution I have no issue with. Look at where they say: "Choose a representation $(n - 3^m)/2 = ...
0
votes
2answers
26 views

How to approach questions that ask to prove a function exists?

Consider the functions $r:S\rightarrow Q$ and $h:S\rightarrow T$ for arbitrary sets $S,T$ and $Q$. Prove that: if $$r(y)=r(x)\Rightarrow h(y)=h(x) $$ then we can find a function $g:Q\rightarrow T$ ...
5
votes
1answer
272 views

$K^n \cong K^m \implies n = m$

Let $K$ be a field and let $E$ be a vector space over $K$. I want to prove that any two finite bases of $E$ are equinumerous. What I did was: Let $B = \{u_1, \cdots, u_n\}$ be a finite basis of $E$. ...
2
votes
2answers
80 views

Prove that even + odd is odd.

Prove that a even number + odd number = odd number Let $x$ be the even number, let $y$ be the odd number. From the definition of odd numbers, $y + 1$ is even. Let: $$x + y = z$$ Suppose $z$ is ...
0
votes
3answers
37 views

Suppose that $f:X \rightarrow Y$ is surjective and $A \subseteq X$ then $f(X-A) \subseteq Y-f(A)$. True or False?

Suppose that $f:X \rightarrow Y$ is surjective and $A \subseteq X$ then $f(X-A) \subseteq Y-f(A)$. I am supposed to determine whether this statement is true or false. If true I am to prove it. If ...
3
votes
5answers
35 views

Prove that $m$ is an integer

Suppose $n$ is a odd integer. It satisfies: $$3^{s} < n < 3^{s+1}$$ For some integer $s \ge 0.$ Show that: $$m = \frac{n - 3^{s}}{2}$$ Is an integer. So, $$2m = n - 3^{s}$$ But that wont ...
1
vote
1answer
75 views

Putnam 2005 A1 Solution [duplicate]

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. (For example, $23=9+8+6.)$ ...
0
votes
0answers
24 views

Seeking help of proving the assertion holds

I am concerned to prove the following assertion holds Noting that $x$ here is just a choice symbol. I have some difficulties to prove it by using Deduction. Any kind of help will be appreciated
-1
votes
3answers
41 views

If A and B are subsets of X then $X -(A-B)= (X- A) \cup B$.

If A and B are subsets of X then $X -(A-B)= (X- A) \cup B$. I think this statement is true. I have attempted to started a proof on it and I am stuck. I am in an introduction to proofs class. ...
0
votes
0answers
58 views

Set theory proof question on rational numbers

I was assigned a problem by my Discrete Mathematics professor that goes as follows: Prove that on $\mathbb{Q}$ (the set of all rational numbers), the relation "$<$" satisfies " $< \circ <~ = ...
0
votes
2answers
33 views

How to apply the mean-value theorem to prove

$f$ is a function that is continuous on $[a, b]$ and differentiable on $(a, b)$. Suppose that there exists $c \in (a, b)$ such that $f(c) > f(a)$ and $f(c) > f(b)$. Prove that there exists $d ...
0
votes
0answers
30 views

Proof: The reduced row echelon form of a matrix is unique.

If $A \in M_{m\times n}$ with real entries, then there exist a unique matrix $R$ in row echelon form such that $A\sim R$, where $R$ comes from $A$ after performing elementary operations. How can I do ...
7
votes
3answers
97 views

Show that if $f : \mathbb{R}^{2} \to \mathbb{R}$ continuously differetiable then $f$ is not inyective

Well my question this time is: How to show that $f : \mathbb{R}^{2} \to \mathbb{R}$ continuously differetiable then $f$ is not inyective I was trying to consider the function $g(x,y)=(f(x,y),y)$, ...
2
votes
1answer
38 views

Proof that v belongs to l_p space under certain conditions

I am struggling with the following problem: Let $M >1$ and $\lambda \in (0,1)$, $\mathbf{z} \in \mathcal{l}_p$. If $|v_t|^p < (M\sum_{s=t}^\infty \lambda^{s-t+1} |z_s| )^p $, then $\mathbf{v} ...
0
votes
4answers
72 views

prove there is no smallest positive rational number

How would I prove there is no smallest positive rational number? what is the best method to prove this statement?
12
votes
6answers
2k views

Prove $1+2\sqrt3$ is not a rational number

How would I go about proving $1+2\sqrt 3$ is not a rational number assuming $\sqrt 3$ is not a rational? Would direct proof be the easiest? Total beginner here, any insight would be appreciated.
1
vote
1answer
35 views

Proving the recursive formula for “Virahanka Numbers”

So apparently Virahanka was an Indian mathematician that, in a way, discovered the Fibonacci sequence 500 years before Fibonacci. He was interested in finding the number of patterns of short syllables ...
5
votes
5answers
106 views

Prove if $n^2$ is even, then $n^2$ is divisible by 4

I am working on this question Prove for every integer n if $n^2$ is even, then $n^2$ is divisible by 4. prove by contradiction Proof: Since there exists an integer $n$ such that $n^2$ is ...
2
votes
0answers
24 views

Help on proving an observation related to pythagorean triangles.

Working for a while on Pythagorean triangles I observed that if $n$ has the prime factorization $p_1^{r_1}p_2^{r^2}...p_i^{r_i}$ where $r_k$'s not all even Then we have: ...
0
votes
1answer
48 views

Linear Algebra Simple Proof

Prove the following result, or provide a counterexample to disprove it. There is a vector space which contains exactly 2 vectors within it. For this question, ...
0
votes
3answers
43 views

Properties of a transpose

If $A$ is a square matrix and $n$ a positive integer, is it true that $(A^n)^T = (A^T)^n$? Justify your answer. Here, I referred to the list of properties for a transpose and there's one that says ...
0
votes
2answers
28 views

Invertible Matrices and Proving The Operations Equal to Identity

Show that if $A$, $B$, and $A + B$ are invertible matrices with the same size, then $$A(A^{–1} + B^{–1})B(A + B)^{–1} = I.$$ This is what i got: $$A^{-1}A+B^{-1}ABA^{-1}+B^{-1}B = I$$ ...
-2
votes
2answers
55 views

The smallest non-abelian group $G$ with a non-normal subgroup [closed]

This time I need to find the smallest non-abelian group $G$ with a non-normal subgroup, then my questions are: 1)Can someone help me to find it? 2)Once we find it, How Can you prove that it is the ...
-1
votes
1answer
24 views

matrix multiplication involving a row of zeros

Show that if A has a row of zeros and B is any matrix for which AB is defined, then AB also has a row of zeros. Here, I know that we can multiply A with B. I also know A has a row of zeros. In matrix ...
0
votes
1answer
46 views

linear algebra properties of the trace of a matrix

Prove: If A and B are n x n matrices, then tr(A + B) = tr(A) + tr(B) I know that A and B are both n x n matrices. That means that no matter what, were always able to add them. Here, we have to do A ...
1
vote
3answers
48 views

How should I write the proof for this situation?

Show that if $A$ is an $m \times n$ matrix and $A(BA)$ is defined, then $B$ is an $n \times m$ matrix. I know that $A$ is a $m \times n$ matrix and to be able to multiply $B$ with $A$, $B$ must be a ...
1
vote
2answers
41 views

simple formal proof $ \lfloor (z+1)/c \rfloor \lt z $ for $ c \ge 2 $

My math is a little rusty. I wish to formally prove to myself that $$\lfloor (z+1)/c \rfloor \le z$$ where $z \in \mathbb{N}$ and $c \ge 2$ for use in a larger inductive proof. The statement seems ...
1
vote
2answers
50 views

Problem in proving that the locus of all points S is a circle.

Given is a circle with midpoint $M$ and a chord $AB$ on this circle. $S$ is the intersection of the altitude from $M$ to $AB$. Prove that the locus of all points $S$ is a circle with midpoint $D$ ...
0
votes
3answers
102 views

$A ⊂ B$ if and only if $A − B = ∅$

I need to prove that $A ⊂ B$ if and only if $A − B = ∅$. I have the following 'proof': $$ A \subset B \iff A - B = \emptyset$$ $$\implies$$ $$\forall x \in A, x \in B$$ Therefore, $$A - B = ...
1
vote
2answers
36 views

Finding an exact sequence and proving that it is split exact

Let $f : M \to M$ be an endomorphism of a module $M$ such that $f^2=f$.Then prove that, $M = \text{Im}\ f \oplus \ker f$. I was thinking that this can probably be proved if I can find an exact ...
1
vote
1answer
41 views

If $f^2$ and $f^3$ are smooth, does it follow that $f$ is smooth?

Let $f: \mathbb{R} \to \mathbb{R}$ be given. Assume that the square and cube of $f$ are smooth. Is $f$ smooth? That is if $f \cdot f \in C^{\infty}$ and $f \cdot f \cdot f \in C^{\infty}$, does it ...
2
votes
1answer
64 views

What is the error in this proof?

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 49, page 583]. This proof seems legit to me. If you know R is transitive, you ...
2
votes
2answers
40 views

If $A \subsetneq C$ then $A \subsetneq B$ or $B \subsetneq C$. Contrapositive?

If $A \subsetneq C$ then $A \subsetneq B$ or $B \subsetneq C$. Is the contrapositive of this statement If $A \subseteq B$ AND $B \subseteq C$ then $A \subseteq C$. I asked because I think the ...
1
vote
2answers
46 views

Why does a set of m elements have 2$^m$ subsets?

Note: This example is from Discrete Mathematics and Its Applications [7th ed, prob 2, pg 576], shout out to @crash. I understand why $A \times A$ has $n^2$ elements(because every member of set $A$ ...
1
vote
4answers
61 views

True or False $A - C = B - C $ if and only if $A \cup C = B \cup C$

True or False $A - C = B - C $ if and only if $A \cup C = B \cup C$ I am in an introduction to proofs class. I think this is a true statement. I have began the proof and realize I have to do this ...
1
vote
1answer
62 views

If $a, b, … e$ $\in \mathbb{R}$ and $a \neq 0$, if $ax + by = c$ has the same solution set of $ax + dy = e$, then these equations are the same

I have an exercise in my last assignment for liner algebra class, where I have to prove that 2 equations are the same. The problem is the following: Prove that, if $a, b, ... e$ are real ...
0
votes
2answers
40 views

Topology proof: dense sets and no trivial intersection

I was wondering if this proof of this basic topological result concerning the closure works. Proposition: Let $A \subseteq (X,\tau)$. Then, $A$ is dense in $X$ if and only if every non-empty open ...
2
votes
3answers
167 views

Proof of the Product Limit Law

Theorem: $$\lim_{x \to a} f(x) = L$$ $$\lim_{x \to a} g(x) = M$$ Then: $$\lim_{x \to a} f(x) g(x) = LM$$ Obviously, $$|f(x) - L| < \epsilon$$ $$|g(x) - M| < \epsilon$$ But multiplying ...
1
vote
1answer
50 views

Irrationality proof trick with Mod [duplicate]

You will see here: Bill Dubuque's Slick $\sqrt{3}$ irrationality proof What is the trick with modulus for proving irrationality? What about $\sqrt{2}$ Can you prove this is irrational by that ...
5
votes
1answer
70 views

need help proving an interval

I am trying to proof $$\frac {1} {ek} \le \frac {1}{k} (1 - \frac {1}{k} )^{k-1} \le \frac {1}{2k} $$ for k>=2 to prove this I first multiply by k getting $$\frac {1} {e} \le \left(1 - \frac ...
0
votes
2answers
38 views

order of subgroup same as order of group(finite groups)

If I have order of a subgroup C of same order as group G I want to prove that G = C. One inclusion is obvious C $\subset$ G the other inclusion we can get by a bijection f : G $\rightarrow$ C hence ...