dbmag9
• Member for 8 years, 9 months
• Last seen this week
• London, UK

As others have said, the question is asking whether it is necessarily the case that whenever $P \Rightarrow Q$ and $Q$ are true, $P$ must be true. My personal favoured instantiations of $P$ and $Q$ ...

As you identify in your question, the real point of contention is the definition of angle. As the other answers have indicated, if your definition includes angles at the intersection of two curves ...

An important point that the other answers haven't (I believe) mentioned: for many mathematical symbols, the natural language 'word-phrase' has a subtly different meaning. A few examples: $\exists$ ...

The derivative $f'(0)$ is defined by $$\lim_{h\to 0} \frac{f(0+h)- f(0)}{h}$$ so you are evaluating $f$ at $x$-values other than $c$. So the question comes down to whether the values of $\frac{f(0+h)-... View answer Accepted answer 2 votes What you are calling the range would more commonly be called the codomain of the function (the range is then defined as the image of the domain, i.e. for$f:A\to B$, the set$\{f(x) : x\in A\}$. With ... View answer Accepted answer 2 votes If you want to distinguish between a set of numbers and a set of representations of numbers, the most obvious thing would be to describe the set as a subset of some well-understood (or at least better-... View answer 2 votes Should throw my mortarboard into the ring and plug Hilary Priestley's Introduction to Complex Analysis. I'm certainly finding it useful. View answer 0 votes Your suggested answer is incorrect. Your condition states that every$x\in X$must appear the same number of times in the first list as it does in the second. So if$(a,b,c)\sim (0,1,2)$then$(a,b,...

Your attempt does not make sense. The biggest problem is your misunderstanding of how the quantifiers work. Expressions like '$\forall x \in \mathbb Z$' and $'\exists x \in \mathbb N$' are not ...
Here's a starting point: Take the well-ordering $\preceq$ on $\mathbb N$ defined by $a\preceq b \text{ if } \begin{cases}\text{$a$is even and$b$is odd}\\a\leq b \text{ where$a$and$b$both odd}... View answer 0 votes Another method, which is no easier than Ángel Mario Gallegos's answer but in another case might well be more elegant: Make the same starting observations as in his: Since$a$,$b$,$c\$ are ...