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 Jan 28 comment Why does $\exists x\,\ x = x$? @CliveNewstead I apologize that my question may have not been clear - the empty set's existence can be proven from the axiom of specification and infinity, it's just that Wikipedia asserted that $\omega$ was not needed, and I was asking why. Jan 28 comment Why does $\exists x\,\ x = x$? @CliveNewstead I found what I was looking for in Andres Caicedo's answer here - all one has to do is negate the existence theorem and prove by contradiction! Jan 28 comment Why does $\exists x\,\ x = x$? @AsafKaragila thanks for the links. I'd argue that this isn't a duplicate however. I see how the existence of the empty set follows from the existence of a set, but my question isn't about the former. Jan 28 comment Why does $\exists x\,\ x = x$? @CliveNewstead I would agree that this would be true if I accept the empty set axiom, but I thought the point was that we didn't need one. Without it, $\exists x\,\ x = x$ wouldn't follow from the axiom $\forall x\,\ x = x$. Jan 28 comment Why does $\exists x\,\ x = x$? @CliveNewstead I don't see why "there exists a set $x$" is meaningless, but, even supposing it is, I still don't see why I can accept $\exists x\,\ x=x$. Jan 28 asked Why does $\exists x\,\ x = x$? Aug 30 awarded Popular Question Aug 22 awarded Yearling Aug 8 comment Is the vector cross product only defined for 3D? I don't understand why we have to prove that a cross product exists for any given $d > r$ - for any set of vectors with size $r$, any matrix which has a column space defined by the basis of this set of vectors is guaranteed to have a rank of at most $r$, and so a null space of $d-r > 0$. Since this null space is always nontrivial, a vector satisfying the criteria for the cross product can always be found. Jul 25 awarded Nice Question Mar 2 answered Is the equality $\frac{d}{dk}\int_{-k}^k \sqrt{f_0(y)f_1(y)}\,\mathrm{d}y=\sqrt{f_0(k)f_1(k)}-\sqrt{f_0(-k)f_1(-k)}$ correct? Dec 4 accepted How did people calculate numerical values of transcendental and trigonometric functions? Dec 4 revised How did people calculate numerical values of transcendental and trigonometric functions? edited title Dec 4 asked How did people calculate numerical values of transcendental and trigonometric functions? Nov 29 answered vector subspaces in $\mathbb{R}^{4}$ Nov 28 suggested rejected edit on Showing $f(0) = 0$ and $|f'(x)| \leq M$ implies $|f(x)| \leq M |x|$. Oct 29 comment Show that the function $g(x)=x^4+x^3+1$ is one-to-one on $[0,2]$ Oh, sorry, never mind. For some reason I interpreted "factor" as "divide out by." My mistake. Oct 29 comment Show that the function $g(x)=x^4+x^3+1$ is one-to-one on $[0,2]$ Why are you allowed to factor out $x_1-x_2$ if they're equal? Oct 27 comment Can you approximate a vector field? It would be one half, because there is an overlap in the information being used for the averages. But the actual implementation of averaging the is immaterial - you could think of lots of ways to do so. Yes, I was referring to a discrete grid of vectors that represents the vector field - that was what your question was asking - how to show general trends - no? Oct 26 comment Can you approximate a vector field? Yes - there would be less information about the field in the image. For example, if you plot some vector field $\vec v$ on a grid with steps $\Delta y$ and $\Delta x$, then after averaging $\vec v'(x, y) = (v(x, y) + v(x+\Delta x, y) + v(x, y+ \Delta y) + v(x-\Delta x, y) + v(x, y- \Delta y))/5$ and plotting a vector on every point on the grid with steps $2\Delta y$ and $2\Delta x$, you'll end up with half the vectors you started with.