Simply decide whether each of the following statements is True or False. Good luck!
If you'd like, you can click this button to reveal how many statements are True and how many are False:
They're all false except for one.
If an open subset of $\mathbb{R}^2$ has a finite perimeter, then its area is also finite.
True
False
The perimeter of $\mathbb{R}^2$ is zero.
If a function $f:\mathbb{R} \to \mathbb{R}$ is smooth and $f(x) \geq 0$ for all $x \in \mathbb{R}$, then there exists a smooth $g:\mathbb{R} \to \mathbb{R}$ such that $g(x)^2 = f(x)$ for all $x \in \mathbb{R}$.
True
False
Take, for $x \neq 0$,
$$f(x) = e^{-1/|x|}\left(\sin^2\left(\frac{\pi}{|x|}\right) + e^{-1/x^2}\right)$$
and $f(0) = 0$. Then $f$ is smooth but $\sqrt{f}$ is only $C^1$ but not $C^2$.
See this paper for more.
Let $a,b,c,d$ be real numbers with $a \leq b$ and $c \leq d$. Then the intervals $(a,b)$ and $(c,d)$ are equal iff $a=c$ and $b=d$.
True
False
The intervals $(2,2)$ and $(3,3)$ are both empty, so they are equal.
Let $(X,d)$ be a metric space and $x \in X$, $r > 0$. Then the closure of $\{y \in X : d(x,y) < r\}$ is $\{y \in X : d(x,y) \leq r\}$.
True
False
Let $X = \mathbb{R}$, $d(x,y) = 1_{x \neq y}$. Then all sets are open and all sets are closed. The closure of $B_d(0,1) = \{0\}$ is itself, whereas $B_d[0,1] = \mathbb{R}$.
Let $X$ and $Y$ be normed spaces and $f:X \to Y$. Suppose that $f$ is linear. Then $f$ is continuous.
True
False
This is a common issue in functional analysis. You can consider, for example, the sequence space $l^\infty_{00}$ of all sequences of real numbers that are eventually 0, equipped with the supremum norm. Take both $X$ and $Y$ to be this space and define $f(\{x_n\}_n) = \{nx_n\}_n$. $f$ is linear, but there is no constant $C$ for which $\|f(\{x_n\}_n)\|_\infty \leq C\|\{x_n\}_n\|$ for all $\{x_n\}_n \in l^\infty_{00}$, so $f$ is not continuous.
Five distinct circles of positive radius in a plane cannot be pairwise tangent.
True
False
For each $r > 0$ let $C_r$ be the circle centered at $(r,0)$ with radius $r$. Then the family of circles $\{C_r\}_{r > 0}$ are all pairwise tangent.
Let $X$ be a compact topological space. A continuous function $f:X \to \mathbb{R}$ must have a maximum and a minimum.
True
False
The empty set topology is compact, but the empty function has no extrema.
The last digit of $11^{10} - 10^{11}$ is 1.
True
False
The quantity is congruent to 1 modulo 10. However, the quantity is negative, so the last digit is technically 9.
Let $N$ be a positive integer, $\emptyset \subset U \subseteq \mathbb{R}^N$ be open and $f:U \to \mathbb{R}$ differentiable. If $\nabla f = 0$ everywhere in $U$ then $f$ is constant.
True
False
One can only conclude that $f$ is constant on each connected component of $U$.
Let $n$ be a positive integer. Then the maximum number of non-overlapping open disks of diameter 1 that can be packed into a $2 \times n$ rectangle is $2n$.
True
False
See Problem 105 in the CMUMC POTD Book.
The groups $(\mathbb{R},+)$ and $(\mathbb{R}^2,+)$ are not isomorphic.
True
False
$\mathbb{R}$ and $\mathbb{R}^2$ are both vector spaces over $\mathbb{Q}$ and they have the same $\mathbb{Q}$-dimension, so they are isomorphic as $\mathbb{Q}$-vector spaces. This implies that they are isomorphic as additive groups.
A rectangular sheet of paper cannot be turned into a (topological) torus by joining its opposite edges without stretching or creasing the paper. (You can consider the paper to have been creased if its deformation is non-differentiable somewhere.)
True
False
There exists a $C^1$ isometric embedding $\mathbb{T}^2 \to \mathbb{R}^3$. See here. (A $C^2$ such embedding does not exist, by appealing to curvature.)
In the empty set topological space (the topological space on $\{\}$ whose only open set is $\{\}$), the empty set $\{\}$ has exactly one open cover.
True
False
It has two open covers: $\{\}$ and $\{\{\}\}$.
There is a unique smooth solution $u(x,t)=u:\mathbb{R} \times [0,\infty) \to \mathbb{R}$ to the heat equation $$\begin{cases}\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0 \\ u(x,0) = 0\end{cases},$$ and it is given by $u(x,t) = 0$.
True
False
There is a pathological solution to the heat equation that satisfies the heat equation by "growing extremely fast". See here. This is why much of the theory concerning the heat equation must be done with a growth assumption on any solution of interest.
Let $f:\mathbb{R} \to \mathbb{R}$ be a non-constant smooth $1$-periodic function. To estimate the integral $I := \int_0^1 f(x)\,dx$, let $L_n$ be the left Riemann sum over the intervals $[0,1/n],[1/n,2/n], \dots, [(n-1)/n, n/n]$, let $R_n$ be the right Riemann sum over those same intervals, and let $T_n$ be the trapezoidal Riemann sum over those same intervals. Then $|T_n-I| < \min(|L_n-I|,|R_n-I|)$ for all $n$ sufficiently large.
True
False
If $f$ is $1$-periodic then $L_n = R_n = T_n$ for all $n$. If it's not immediate, try writing out the expressions for each of these Riemann sums and combine like terms.
Let $R$ be a symmetric and transitive relation. Then $R$ is reflexive.
True
False
The empty relation is not reflexive.
Let $A$ and $B$ be open non-empty convex subsets of $\mathbb{R}^2$ whose boundaries are smooth. Then their Minkowski sum $A+B$ is an open convex set with smooth boundary.
True
False
This is a real doozy. Please see the paper Smoothness of Vector Sums of Plane Convex Sets by Kiselman. It turns out that the maximum regularity that can be guaranteed of the boundary of $A+B$ is $C^{20/3} := C^{6, 2/3}$. Example 2.2 of the aforementioned paper obtains this maximum regularity: Take $A = \{(x,y) : y > x^4/4\}$ and $B = \{(x,y) : y > x^6/6\}$.
For a finite non-empty set $S$, denote by $\mu(S)$ the average of $S$. Let $A_1,A_2,B_1,B_2$ be finite, non-empty and pairwise disjoint sets of real numbers. If $\mu(A_1) > \mu(B_1)$ and $\mu(A_2) > \mu(B_2)$ then $\mu(A_1 \cup A_2) > \mu(B_1 \cup B_2)$.
True
False
This is Simpson's Paradox. Here is an example (using multisets instead of sets): Take $A_1 = \{0,0\}$, $B_1 = \{0\}$, $A_2 = \{1000\}$, $B_2 = \{1000, 1000\}$. Then $\mu(A_1) \geq \mu(B_1)$ and $\mu(A_2) \geq \mu(B_2)$, but the average of $A_1 \cup A_2$ is $\approx 333$ whereas the average of $B_1 \cup B_2$ is $\approx 666$. One can easily perturb the numbers to more precisely contradict the original statement.
The graph of any continuous function $f:[a,b] \to \mathbb{R}$ can be drawn by tracing it with a pen at constant speed, starting at $(a,f(a))$ and ending at $(b, f(b))$, without ever picking up the pen.
True
False
Take $a=-1$, $b=1$, and $f(x) = x\sin(1/x)$ with $f(0) = 0$. You can show that the arc length of the graph of $f$ over $[-1,1]$ is infinite (for example, you can show that $f$ is not of bounded variation, which is sufficient).
Any polygon (in $\mathbb{R}^2$) can be continuously deformed into a convex polygon without bending its sides, changing the lengths of its sides, or having any two sides' interiors intersect at any time.
True
False
The answer is yes, but it is highly non-trivial. This is known as the Carpenter's Rule problem. It is natural to consider the same question of other dimensions: Suppose we were to link together a number of rigid ``rods" in $\mathbb{R}^n$, attached at their endpoints (but free to pivot around these endpoints), to form a polygon (a loop of such rods). Can the rods be moved to form a planar convex polygon without having any two rods pass through each other? The answer is yes for $n=2$ and $n \geq 4$. It is false for $n = 3$. You can find a counterexample here.