5
votes
Is this proof about independence of events rigorous?
This is not a proof but rather an intuitive explanation of what is happening behind the curtains. While the reasoning is sound, it does not prove the statement as it lacks the unambiguous formalism to ...
- 8,011
4
votes
Accepted
Exercise 3.6.8 from "How to prove it"
For your $A \subseteq U$, define $B = U \setminus A$. Then for any $C \subseteq A$:
$$
C \cap B = C \cap (U \setminus A) = (C \cap U) \setminus (C \cap A) = C \setminus (C \cap A) = C \setminus A
$$ ...
- 6,642
4
votes
Exercise 3.6.8 from "How to prove it"
You're unlikely to be able to complete the proof if you have no idea what $B$ should be. At this point (or even before you start trying to write the proof), consider some examples, try to visualize ...
- 9,337
4
votes
Accepted
Can we use Prove by Contradiction to prove a statement FALSE?
That works, because if you have a statement $P$ and you want to show $P$ is false, you can do proof by contradiction with the statement $Q=\neg P$. You deduce $Q$ is true, so you deduce $P$ is false. (...
- 25.7k
4
votes
Prove that $x^3 = x^2$ has exactly two solutions
On one hand we know that $1^3=1^2$ and $0^3=0^2$ so $0$ and $1$ are solutions.
On the other hand $$x^3=x^2\implies x^3-x^2=0\implies x^2(x-1)=0$$
Therefore, if $x$ is a solution then $x$ must be ...
- 10.3k
3
votes
Accepted
Prove that $x^3 = x^2$ has exactly two solutions
It obviously proves that $z=1$ is possible, but how does it prove that
$z=0$ is possible?
No, that is not what the proof you cite is proving. The proof is not saying that $z=1$ is possible, nor is it ...
- 120k
2
votes
Suppose $A$ and $B$ are two sets with $B\subset A.$ Let $f:A\to B$ be injective. Then show that $\exists$ a bijection $h:A\to B.$
The above alternative proof is justified/valid only under the assumption, that we are working with finite sets.
To elaborate, the proof might be true in general, but it's not a complete one. It is ...
- 411
2
votes
Accepted
How to quote definitions concisely?
The negative of any even integer is even.
Correctly applying a definition does not necessarily involve quoting the thing. For example, here's a sample proof:
Let $n$ be an arbitrary even integer. ...
- 36k
2
votes
Accepted
How to know if a variable is an integer when writing proof?
An integer multiplied by an integer, is an integer. This is because the set of integers is closed under multiplication. In your case, we have $2r^2$, because $r$ is an integer we get that $rr=r^2$ is ...
- 98
2
votes
Proof verification for $(2n)! ≥ 2^n (n!)^2$
I would do it like this:
$$(2n)!\ge2^n(n!)^2$$
$$\frac{(2n)!}{n!}\ge2^n\cdot n!$$
Notice that...
$\frac{(2n)!}{n!}=\color{#AA0000}{2n}\color{#AA7700}{(2n-1)}\color{#AAAA00}{(2n-2)}\dots\color{#77AA00}...
2
votes
Accepted
Showing that $\{\varphi_{m,n} \}_{m \geq 1, n \geq 1}$ is an orthonormal basis for $L^2((a,b) \times (a,b)).$
Apply Fubini/Tonelli's Theorem to $g(s,t)=\chi_{\{(s,t): f(s,t) \neq 0\}}$. Since $\int g(s,t) ds=0$ for almost all $t$, we get $\int \int g(s,t) dsdt=0$ by integration w.r.t. $t$.
- 24k
2
votes
Accepted
Is this a valid proof by contradiction for why there are infinitely many primes?
This is because I am unsure whether it is correct for us to identify a greatest prime number P and then construct an even greater prime number Q right afterwards- this feels like it was not legitimate ...
- 36k
2
votes
How well should I know understand the theorem before proving it?
Some theorems have profound proofs, and require developing a clever idea. And some theorems follow very directly from definitions, and their proofs are basically bookkeeping*. Your example seems to be ...
1
vote
Accepted
If $X=\{0,1\}$ then for any set $A,$ prove that $|X^A|= |P(A)|.$
The red step is where your proof breaks down:
$$X^A=|X|^{|A|}=2^{|A|} \color{red}{= |P(A)|}$$
The first two steps are fine (your lemma shows the first equality and the second equality holds by ...
- 3,339
1
vote
Explicit formula for Hermite polynomials.
Due to the nature of the exponential function we know that for all $n\in\mathbb N_0$ there exists a polynomial $p_n:\mathbb R\to\mathbb R$ such that:
$$
\frac{d^n}{dt^n}e^{-t^2}=p_n(t)e^{-t^2}
$$
...
- 4,111
1
vote
Prove that $x^3 = x^2$ has exactly two solutions
Well, we can do it like this:
$$x^3=x^2$$
$$x^2\cdot x=x^2\cdot1$$
Now, dividing both sides by $x^2$, we get that $x=1$ is one solution. But how is $x=0$ also a solution? 🤔
Aha! We can't divide by $0$...
1
vote
Prove that $x^3 = x^2$ has exactly two solutions
$\newcommand{\fitch}[2]{\begin{array}{|l}#1\\\hline#2\end{array}}$
The uniqueness part is just showing $\lnot\exists c\!\in\!\mathbb{R} (c^3 = c^2 \land c\neq 0 \land c\neq 1)$. This is just some ...
- 2,129
1
vote
Accepted
To prove $((\varphi \to \psi) \land \chi) \implies \sigma$, which formulas can I consider true?
The formula $$\big((P\to Q) \land R\big) \to S$$ is equivalent to each of the following
$\Big((\lnot P\land R) \to S \Big) \quad\land\quad \Big((Q\land R) \to S\Big)$
$R\to \Big( (\lnot P \to S ) \...
- 36k
1
vote
Is a construction/definition a premise in a logical argument that a proof is?
In general, you can think of the "premises" of the proof to be solid and unchangeable facts that can be used as necessary. However, more of these "premises" can be constructed as ...
- 33
1
vote
To prove that for every $x$, $(x\in Z\implies x\in R),$ is it ok to write "For any $x,$ suppose $x\in Z$. Then... Then $x\in R$"?
Your proof is fine.
For a slightly more precise wording I would recommend starting with
Let $(x,y)\in ...$
and then proceeding exactly as you have.
NOTE
Despite some of the comments about "let&...
- 1,301
1
vote
Disproof: there exists an integer $k ≥ 4$ such that $2k^2 − 5k + 2$ is prime.
Prove the following claim is true: "$P$"
Prove by contradiction, assume "$\neg P$" is true.
Prove the following claim is false: "$Q$"
Prove by contradiction, assume "...
- 15.4k
1
vote
Accepted
Need a proof for a lemma (Stewart. Galois Theory. Lemma 5.14, about the dimension of simple extensions)
I'm not sure either what they meant by a restatement of lemma $5.9$. Let's look at lemma $5.14$. Consider the minimal polynomial of $\alpha$ over $K$, denoted by $f^{\alpha}_K := f = \sum^n_{i=0} b_i ...
- 245
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