This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

# Questions tagged [proof-explanation]

10133 questions

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### Not understanding the proof that there is no surjection from a set to its powerset

Here is the question:
If a set, $A$, is finite, then $|A| < 2^{|A|} = |P(A)|$, and so there is no surjection from set $A$ to its powerset. Show that this is still true if $A$ is infinite.
Here is the proof:
We prove there is no surjection by…

Dak Song

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### Help needed in proof that $\sqrt2$ is irrational

In the following proof that $\sqrt2$ is irrational, I cannot make sense of why $\frac{q a_n + p b_n}{q} \geqslant \frac{1}{q}$.
The sums corresponding to $a_n$ and $b_n$ are alternating sums, so why must $q a_n + p b_n \geqslant 1$…

HoopaU

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### Where does the premise of this idea come from?

Let $x$ , $y$ be positive real numbers. Prove the inequality $$x^ y + y^x \ge 1$$
This is the solution provided by my textbook:
Where does this first idea (proving that $a^b \ge \frac{a}{a+ b - ab}$) come from? I'm repeatedly frustrated by…

Airdish

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### Perfectly Normal is hereditary

The definitions I'm working with:
$(X, T )$ is called perfectly normal if whenever $C$ and $D$ are disjoint, nonempty, closed subsets of $X$, there exists a continuous function $f : X \rightarrow [0, 1]$ s.t $C = f^{−1}(\{0\})$ and $D =…

ಠ_ಠ

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### If $G$ is a graph of order $n$ such that $\delta (G) ≥ (n-1)/2$ , then $\lambda(G) = \delta(G)$

Prove that if G is a graph of order n such that δ(G) ≥ (n-1)/2 , then λ(G) = δ(G).
where
δ(G)= minimum degree of the graph G
λ(G)= minimum edge cuts to disconnect graph G
κ(G)= minimum vertex cuts to disconnect graph G
I know by a theorem that…

Jason

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### Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers

Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers
It is in the exercises of the AM-GM inequality chapter of a book, and that is why I believe it will be solved by that. Can anyone give me a proof using that or…

VSA

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### Help understanding proof of Frostman's Lemma - issue technical or termonological?

I was reading Hochman's proof of Frostman's lemma in his online lecture notes here and got hung up. I'm not sure if I'm missing a part of the proof or I'm misunderstanding the theorem itself. The theorem (Theorem 4.11, equivalent to Theorem 4.14)…

AJY

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### Principle of Mathematical induction proof

Prove that $2^n >n$ for all positive integer $n.$
I know this can be easily proved by using PMI
Let $P(n): 2^n > n$
For $n = 1$ $$2^1 > 1.$$ Hence $P(1)$ is true.
Assuming that $P(k)$ is true for any positive integer $k$ i.e.
$$P(K) = 2^k >…

Heisenberg

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### Groups and multiply both sides.

I'm just starting to learn group theory by reading "Abstract Algebra" by Dummit and Foote. Apologies if this question is stupid, I don't have a very strong background in mathematics. I'm trying to prove the very early propositions to myself…

Necrototem

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### Prove: if $f:\mathbb{N} \rightarrow A, g:\mathbb{N} \rightarrow B$ are surjections, there exists a surjection $h:\mathbb{N} \rightarrow A \cup B$

I chose my own sets here for A and B as countably infinite pairwise disjoint subsets of $\mathbb{N}$. Can I do this with finite subsets and get an easier answer with the same result?
Suppose $A = \mathbb{N}_e$ such that $\mathbb{N}_e$ is the set of…

J00S

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### How can ($A$ and $B$ $\implies$ $C$) and ($C$ and $B$ $\implies$ not $A$) together imply (not $A\iff B$)?

I encountered this two statements when I tried to understand the proof of Kuratowski Theorem.
Any minimal nonplanar graph and it has no Kuratowski subgraphs, then it must be at least 3 connected.
An at least 3 connected graph and it has no…

user338393

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### For $f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(x)=ax+b$, for what $a,b \in \mathbb{R}$ is $f$ a bijection?

I asked a similar question here.
This question has different parameters however as you can see.
For $f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(x)=ax+b$, for what $a,b \in \mathbb{R}$ is $f$ a bijection?
Observe the case where $a = 1$ and $b…

J00S

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### Need hint on induction proof for summation

I have a homework problem to prove the following via induction:
$$\sum_{i=1}^n i^22^{n-i} = 2^{n+3}-2^{n+1}-n^2-4n -6$$
The base case is true. I generated the below using $s_k+a_{k+1}=s_{k+1}$:
$$…

Chase Street Dev

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### Conditional Radon-Nikodym and disintegration

Here (p. 15) the author defines conditional divergence as
$$D(P_{Y\mid X}\mid\mid Q_{Y\mid X}\mid P_X):=\mathbb{E}_{x\sim P_X}\left[D(P_{Y\mid X=x}\mid\mid Q_{Y\mid X=x})\right]$$
for two conditional distributions $P_{Y\mid X}$ and $Q_{Y\mid X}$ and…

Stefan

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### If $f: X \rightarrow Y$ is continuous, and $X$ is connected, then $f(X)$ is connected

I'm trying to understand the proof of:
If $f: X \rightarrow Y$ is continuous, and $X$ is connected, then $f(X)$ is connected.
What are we trying to do in the following proof (are we proving the contrapositive or trying to prove by…

Irregular User

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