All Questions
Tagged with natural-numbers proof-explanation
19
questions
3
votes
5
answers
333
views
Are there nonzero natural numbers such that $\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}$?
Check if there are nonzero natural numbers $n,x,y$ such that:
$$\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}. $$Thank you in advance!
My ideas
So we can simply show that $4n+5,5n+1,9n+4$ are ...
- 748
0
votes
1
answer
163
views
Proving (rigorously) that the number of $m$ element subsets of an $n$ element set is ${n \choose m}$
I am trying to solve the following problem (Amann & Escher Analysis I, Exercise I.6.3):
Show that the number of $m$ element subsets of an $n$ element set is ${n \choose m}$.
I emphasize that the ...
- 1,143
0
votes
1
answer
193
views
Prove without the well ordering principle that no m exists such that $n < m < n + 1$ for positive integers $m$ and $n$. [duplicate]
I'm trying to prove without the well ordering principle that no integer $m$ exists such that $n < m < n + 1$ for positive integers $m$ and $n$.
I know there's a proof here that uses the well ...
- 383
1
vote
2
answers
145
views
Well Ordering implies Induction Proof doubt
I’m trying to understand the proof for the fact that that the Principle of Well-Ordering implies the Principle of Mathematical Induction; that is, if S ⊂ N such that 1 ∈ S and n + 1 ∈ S whenever n ∈ S,...
- 143
1
vote
1
answer
709
views
How does this prove that the set of positive integers is unbounded above
This is a proof from Apostol's Calculus. What I don't understand about it is the "Hence, there is at least one positive integer n such that n > b - 1" Is it because for every inductive ...
3
votes
1
answer
364
views
Showing that the natural numbers are totally ordered with respect to set membership
Working with the usual set theoretic construction of the natural numbers, denoted $\omega$ for now.
I am trying to show that $\omega$ is totally ordered with respect to set membership, that is, $n<...
- 3,612
0
votes
3
answers
82
views
I am trying to prove gcd$(8,20) = 4$
I am trying to prove $\gcd(8,20) = 4$.
I am not quite sure how to go about it, but I think I need to prove that $\gcd(8,20)$ is not equal to $1$.
I have also set up $S = \{k \in \mathbb{N} : k = 8x + ...
- 133
0
votes
2
answers
54
views
How to prove that for any two natural numbers a, b, there exist natural numbers c,d with a + d = b +c
I understand the intuition for how to do this, but I don't know how to formally prove it. For example, I know that if you take (a,b) to be (1,2), then a sample (c,d) could be (1,2) since 1 + 2 = 2 + 1....
2
votes
1
answer
40
views
For all $a,b,n \in \mathbb N$, $0 \leq n(a+b+1)-n^2 +b$
I claim the following:
For all $a,b,n \in \mathbb N$, we have that $0 \leq n(a+b+1)-n^2 +b$.
This seems true for me... Although how do I really check if this is true? I was thinking about doing it ...
- 3,794
0
votes
1
answer
34
views
Examples of basis
Excuse me , can you see this question,
For each positive integer $n$ , let $S_n=\{n,n+1,\ldots\}$ . The collection of all subsets of natural number which contain some $S_n$ is a base for a topology on ...
- 93
1
vote
1
answer
26
views
Prove that $ \text{Inc}(n,m) = \text{Inc}(m-1, m) \circ \text{Inc}(n, m-1)$
Let $n, m \in \mathbb N$ and also let $n \leq m$, we define the set: $$\text{Inc}(n,m): = \{f: \{1,...,n\} \to \{1,...,m\} \mid f \text{ strictly increasing}\}.$$
I am trying to prove (for $n<m$ I ...
user459879
1
vote
0
answers
53
views
PROOF: A Relationship Between A Natural Number and The Quantity of Its Divisors' Divisors
An arbitrary natural number $N \in \mathbb{N}$ has the divisors $d_1,d_2,...,d_s$ , where $N$ has $s$ divisors in total. If $n_i$ denotes the number of dividers to $d_i, i = 1, ..., s$, it can be ...
user736333
1
vote
0
answers
502
views
Well Ordering Principle sum of natural Numbers help
I was reading over the following solution for the sum of all natural numbers using the well-ordering principle.
Let Fact 1 be:
$$\forall n:\ (1+2+3+...n)=\frac{n(n+1)}{2}$$
By contradiction, ...
- 117
0
votes
1
answer
290
views
Proving commutativity of multiplication
I am trying to prove Lemma 2.3.2 in Tao's analysis text: that for any two natural number, $n$ and $m$, we have $n \times m = m \times n$. I only have the properties of the natural numbers and addition ...
user465188
0
votes
1
answer
158
views
Proof that $a < b$ if and only if $a{+\!+} ≤ b$
I am trying to prove part (e) of Proposition $2.2.12$ in Tao's analysis textbook that for natural numbers $a, b$, $a < b$ if and only if $a{+\!+} ≤ b$, where $a{+\!+}$ is the successor of $a$. I am ...
user465188