Comment by CompuChip on Does continuity depend on the distance function?
Indeed if it weren't we wouldn't have to define continuity for functions between metric spaces -- we could just define it as a property of sets $X$ and $Y$. If you think about the intuitive meaning of...
View ArticleComment by CompuChip on Zero/Zero questions and perhaps faulty logic
"you have to check whether a is zero before you cancel terms off" - otherwise you get such interesting mistakes like: "Solve x² = x" --"Cancel x on both sides ==> x = 1; done!".
View ArticleComment by CompuChip on Definition of transitivity.
"To paraphrase Anouk, $y$ is nobody's wife." - Actually, $y$ is $x$'s wife... and I'm not sure "nobody is $y$'s wife" is a song. The whole thing is a stone hanging 'round 123's neck though :)
View ArticleComment by CompuChip on Isometry of two metric spaces
Easy: just prove that all norms on $\mathbb R^2$ are equivalent,then this follows trivially.
View ArticleComment by CompuChip on Probability of getting heads given that first flip...
One weird remark about conditional probabilities here: @JoeTaxpayer would be completely right a priori. You could decide to flip a coin 50 times, and call it unfair if you get more than, say, 35 heads,...
View ArticleComment by CompuChip on Proof by contrapositive: if $a$ and $b$ are...
If you want to do more work than necessary, don't use contrapositive but prove (1) even + odd = odd and (2) if a, b are consecutive then one of them is even and the other is odd.
View ArticleComment by CompuChip on Why is the derivative important?
Hmm if there is any part of mathematics that has practical applications, I would say it is derivatives. What world are your professors living in that they cannot give you any useful example?
View ArticleComment by CompuChip on A fly has a lifespan of $4$–$6$ days. What is the...
Does anyone else find the phrasing confusing? "The fly will die at midnight on the fifth day" clearly refers to one time instant and then the reasoning given is correct, but initially I read "exactly...
View ArticleComment by CompuChip on Construct a function which increases faster than its...
"Then there is some interval (1−ϵ,ϵ) on which log(f(r)) is defined" -- did you mean: "some interval $(1 - \epsilon, 1)$" ?
View ArticleComment by CompuChip on Help to identify every equation in this meme?
+1 for identifying the textbook!
View ArticleComment by CompuChip on What is the probability of a card of heart or non...
So the allowed cards are all the hearts, and all non-jacks? That is, all the cards which are not jacks, and the jack of hearts? Because those are indeed 49 card.s
View ArticleComment by CompuChip on Show that if $(z+1)^{100} = (z-1)^{100}$, then $z$ is...
So it is true for all $z \in i \mathbb R$?
View ArticleComment by CompuChip on Is a vector space a subspace of itself?
@TheoBendit for sets, a proper subset is neither the empty set nor the entire set. So for vector spaces, isn't the trivial ({0}) space also excluded as a proper subspace?
View ArticleAnswer by CompuChip for Validity of notation from the aspect of function...
You are not using $a$ at all. Also, your restriction on $A$ should be quantified. Did you mean to start with $\forall A \subset \mathbb{N} \cdots$ ?You are slightly mixing what you are defining and...
View ArticleAnswer by CompuChip for Shell method for calculating volume of solid of...
OK, here's as far as I got. It didn't quite work out yet, I think I made a calculation error somewhere, but I'm posting it as an answer because I'm confident that you'll be able to work out the...
View ArticleAnswer by CompuChip for A schoolbook question about an inequality equivalence.
Suppose we accept the implication you quoted:$$\text{for any }a, b \in \Bbb R\text{, }\quad \forall c \in \Bbb R, \quad a > b \implies a + c > b + c \quad \qquad (\ast)$$and try to prove the...
View ArticleAnswer by CompuChip for Show that (p ∧ q) → (p ∨ q) is a tautology?
I know you asked specifically about a given proof, but here is another way:(1) Assume p ∧ q(2) By ∧-elimination, p(3) By ∨-introduction, p ∨ q(4) By →-introduction and marking the assumption (1), (p ∧...
View ArticleAnswer by CompuChip for Probability of being poisoned
The other answers are great, but with a slightly different approach from your attempt (I presume you edited in your attempts after some of the answers are posted). I'd like to point out some important...
View ArticleAnswer by CompuChip for Probability of winning in a die rolling game with six...
I also got to 2/7, but in a different way.Let $p$ be the probability that player 1 wins the game, either on his current turn or in the future. Let $q$ be the probability that player 1 eventually wins...
View ArticleAnswer by CompuChip for Is the Pi in reversed order Real number?
Let $(a_n)_{n \ge 0}$ be the sequence of digits of $\pi$: $a_0 = 3$, $a_1 = 1$, $a_2 = 4$, $a_3 = 1$, etc such that $$\pi = \sum_{k = 0}^\infty a_k 10^{-k}.$$Now define the sequence$$r_n = \frac{1}{10}...
View ArticleAnswer by CompuChip for How can mathematical induction prove something?
So suppose that you have some theorem, and you have followed the two steps (check the base case, and did the inductive step). Now I doubt whether the theorem is actually true for $n = 6$. How can you...
View ArticleAnswer by CompuChip for Cant figure out how to solve this probabilty problem?
Although mathbeing's answer is correct, you wanted to identify the sample space so I'll help you with that.There are probably different ways of setting this up, but for me the most logical way (most...
View ArticleAnswer by CompuChip for How do we prove that something is unprovable?
You have already received some excellent responses, but I just wanted to clarify a few things that I feel may have not received the attention that they require.For instance, you mentioned[...]...
View ArticleAnswer by CompuChip for Is this set of vectors linearly (in)dependent?
Although other answers have pointed out the fast track (more than $n$ vectors cannot be linearly independent in an $n$-dimensional space), I just wanted to help you complete your own approach, which...
View ArticleAnswer by CompuChip for $( \ X\to G \ , \ \star \ )$ is a group if $(\varphi...
You can actually work out what an inverse looks like quite explicitly. Take an arbitary $\varphi \in \mathcal X$. The question is, can we construct a function (let's denote it, completely...
View ArticleAnswer by CompuChip for One definite integral-multiple answers?
You will have to decide whether to integrate over $x$ or $x^2 + 1$, you can't pick and choose.To make this more clear, let's do a proper substitution and define $u = 1 + x^2$. Then $\mathrm du = 2x \,...
View ArticleAnswer by CompuChip for How do I show that the sum of the reciprocals of...
The other answers give you tighter estimates, I think, but since your upper bound is quite relaxed (the sum you're looking for is actually around 1.2) let's have some fun with the fact...
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