# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### Suppose f : A ---> B and g : B ---> A are functions for which g o f = 1A…

If I were to suppose that $f : A \to B$ and $g : B \to A$ are functions for which $g \circ f = 1_A$, is $f$ always surjective and is $g$ always injective? How would I either prove this or counter it? ...
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### Having trouble with showing this CANNOT be a theorem in incidence geometry.

Consider the following statement: If l and m are any two distinct lines, then there exists a point P that does not lie on either l or m. (a) Show that this cannot be a theorem in incidence geometry. ...
256 views

### Verify the following combinatorial identity: $\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$ [duplicate]

$$\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$$ Nice, so I've proven some combinatorial identities before via induction, other more simple ones by committee selection models.... But ...
122 views

### About using Archimedean property to prove the existence of least upper bound

I am studying the proof of existence of least upper bound, but I can not understand how the autor applies the Archimedean property. Archimedean property. Let $x$ and $\epsilon$ be any positive ...
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### Prove that the kernel is of dimension 2

"Experimentally", I found that the kernel (null space) of the following matrix is of dimension 2. I'd like to prove it, but haven't managed yet: \text{for almost all } t>0,\quad ...
33 views

### Prove If LCM(a,b) = c and a|k and b|k then c|k.

Prove If LCM(a,b) = c and a|k and b|k then c|k. I know that c divides a and b if c = Least Common Multiple of a and b. I also know that c divides all multiples of a and b. I just am not sure how ...
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### Prove this sum of binomial terms using induction.

Here's the problem stumping me today: Let $n \in \mathbb{N}$ and $r \in \mathbb{N}$ such that $r \leq n$, and prove using induction that $\binom{n+1}{r+1} = \sum\limits_{i=r}^n \binom{i}{r}$. I've ...
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### Suppose that the $\lim a_{n} = 1$ and $x < y$

Suppose that $\lim a_{n} = 1$ and $x<y$. Is it possible to show using the Algebriac limit theorem that if the $\lim xa_{n} < y$. Then $0<x<y$? .
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### no. of regions a plane is divided into by $n$ lines in general position

My notes state the Counting process for knowing no. of regions a plane is divided into by $n$ lines in general position := Let $h_1(n)=$ No. of parts a line is divided by $n$ distinct ...
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### Prove [(xy=0)∧(x,y∈ℤ)]→[(x=0)∨(y=0)]

I'm working through a higher algebra textbook. It has some exercises related to the positive integers and I'm stuck on this proof. Here's what I have so far: Attempted proof Assume the contrary, ...
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### $[x]_0,[x]_1\ldots [x]_n$ is a basis for vector space $V$.

here is a lemma which requires the use of falling factorials which are written as $[x]_n=x(x-1)\ldots(x-(n-1))$ : Lemma:Let $V$ be a vector space of polynomials over $\mathbb C$ , then ...
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### show that if det()=0 with A, and B, then AB = BA.

show that if |b c| |a b| = 0 with A = [a a] [b b] , and B = [b b] [c c], then AB = BA. I'm not exactly sure how to go about proving this. I tried computing both AB and BA, then factoring ...
58 views