TsinghuaX Calculus-1 Notes: Chapter 1 Numbers and Sets (Unit 1-3)

Unit 1 Review of Real Numbers [Lecture PDF]

1. Which one of the following equations is always TRUE?
A. ${(p+q)}^2=p^2+q^2$
B. $\frac{1}{x-y}=\frac{1}{x}-\frac{1}{y}$
C. $\frac{1/x}{a/x-b/x}=\frac{1}{a-b}$
D. $\frac{1+xy}{y}=1+x$

2. Which of the following statements are NOT NECESSARILY TRUE?
A. If $a\in\mathbb{N},\ b\in\mathbb{Z^+}$, then $a+b\in\mathbb{N}$.
B. If $a\in\mathbb{N},\ b\in\mathbb{Z^+}$, then $ab\in\mathbb{N}$.
C. If $a\in\mathbb{Q},\ b\in\mathbb{Q}$, then $ab\in\mathbb{Q}$.
D. If $a\in\mathbb{Q},\ b\in\mathbb{Q}$, then $\frac{a}{b}\in\mathbb{Q}$.
E. If $a\in\mathbb{Q^c},\ b\in\mathbb{Q^c}$, then $ab\in\mathbb{Q^c}$.

Solutions:
1. C is correct.
${(p+q)}^2=p^2+q^2+2pq$, $\frac{1}{x}-\frac{1}{y}=\frac{y-x}{xy}\neq\frac{1}{x-y}$, $\frac{1+xy}{y}=\frac{1}{y}+x$.
2. B, D, E are correct.
$\mathbb{N}$ is natural numbers, e.g. $\{1, 2, 3,\ldots\}$.
$\mathbb{Z^+}$ is non-negative integers, e.g. $\{0, 1, 2,\ldots\}$.
$\mathbb{Q}$ is rational numbers, and $\mathbb{Q^c}$ is irrational numbers. $\mathbb{Q}+\mathbb{Q^c}=\mathbb{R}$.


Unit 2 Exponentiations, Logarithms and Sets [Lecture PDF]

1. Simplify the expression: $$({\frac{3x^{3/2}y^3}{x^2y^{-1/2}})^{-2}}$$
2. Choose the CORRECT one without using a calculator.
A. $e^{\ln{2}}\lt e$
B. $e^{\ln{\pi}}\lt e$
C. $e^{\lg{1000}}= 3$
D. $(\lg{24})\times(\lg{25})=\lg{49}$

3. Let $A=\{a|a=n^2+1,n\in\mathbb{N}\},\ B=\{b|b=k^2-4k+5,k\in\mathbb{N}\}$ be two sets, then
A. $A=B$
B. $A\subseteq B$
C. $B\subseteq A$
D. None of the above is true

Solutions:
1. $({\frac{3x^{3/2}y^3}{x^2y^{-1/2}})^{-2}}=\frac19 xy^{-7}$
2. A is correct.
$2\lt e\lt3\lt\pi\Rightarrow \ln2\lt1\lt\ln3\lt\ln\pi\Rightarrow e^{\ln{2}}\lt e\lt e^{\ln\pi}$.
$e^{\lg{1000}}= e^3 > 2^3>3$
$\lg({24\times25})=\lg{24}+\lg{25}$
3. B is correct.
$a=n^2+1\ge2$ and $b=k^2-4k+5=(k-2)^2+1\ge1$.


Unit 3 Limit Points, Open and Closed Sets [Lecture PDF]

1. Here are some sets:
(1) $\mathbb{R}$
(2) $\varnothing$
(3) $(1,+\infty)$
(4) $[-1,0]$
(5) $\{1,2,3\}$
(6) $\{y|y=2x^2+1,x\in [0,2)\}$
(7) $\mathbb{Q}$
(8) $\mathbb{Q^c}$
Among the above sets, the total number of open set is? the total number of closed set is?

2. Consider the set $S=[1,2)\bigcup \{0\}$, Which of the following statements about $S$ are TRUE?
A. $x=1$ is not an interior point of $S$
B. $x=0$ is not an interior point of $S$
C. $x=0$ is not a limit point of $S$
D. $x=2$ is not a limit point of $S$

Solutions:
1. 3 open sets and 4 closed sets.
$\mathbb{R}$ and $\varnothing$ are either open or closed sets;
$(1, +\infty)$ is open;
$[-1, 0]$ is closed;
$\{1, 2, 3\}$ is closed (finite sets are closed);
$\{y|y=2x^2+1,x\in [0,2)\}=[1,9)$ is neither open nor closed;
$\mathbb{Q}$ and $\mathbb{Q^c}$ are neither open nor closed.
Therefore, there are 3 open sets and 4 closed sets.
2. A, B, C are correct.
$x=1$ is a limit point, is not an interior point;
$x=0$ is neither a limit point, nor an interior point;
$x=2$ is either a limit point or an interior point;
Any other points in S are limit points, as well as interior points.


Some Key Notes of Week-1 (Chap.1 Uni.3) [Reference Link]
Open and Closed sets

1. A set $\mathbf{U}\subset\mathbb{R}$ is called open, if for each $x\in\mathbf{U}$ there exists and $\varepsilon>0$ such that the interval $(x-\varepsilon, x+\varepsilon)$ is contained in $\mathbf{U}$. Such an interval is often called an $\varepsilon$ - neighborhood of $x$, or simply a neighborhood of $x$.
2. A set $\mathbf{F}$ is called closed if the complement of $\mathbf{F}$, $\mathbb{R}\setminus\mathbf{F}$, is open.

Basic Examples:
$(a, b)$ is open.
$[a, b]$ is closed because its complement sets $(-\infty, a)\cup(b, +\infty)$ are open.
Both $(a, b]$ and $[a, b)$ are neither open nor closed.
Both $(0, +\infty)$ and $(-\infty, 0)$ are open.
Both $[0, +\infty)$ and $(-\infty, 0]$ are closed.
$\mathbb{R}$ and $\varnothing$ are either open or closed.

Boundary, Accumulation, Interior, and Isolated Points
Let $\mathbf{S}$ be an arbitrary set in the real line $\mathbb{R}$.

1. A point $b\in\mathbb{R}$ is called boundary point of $\mathbf{S}$ if every non-empty neighborhood of $b$ intersects $\mathbf{S}$ and the complement of $\mathbf{S}$. The set of all boundary points of $\mathbf{S}$ is called the boundary of $\mathbf{S}$.
2. A point $s\in\mathbf{S}$ is called interior point of $\mathbf{S}$ if there exists a neighborhood of $s$ completely contained in $\mathbf{S}$. The set of all interior points of $\mathbf{S}$ is called the interior.
3. A point $t\in\mathbf{S}$ is called isolated point of $\mathbf{S}$ if there exists a neighborhood $\mathbf{U}$ of $t$ such that $\mathbf{U}\cap\mathbf{S}=\{t\}$.
4. A point $r\in\mathbf{S}$ is called accumulation point (i.e. limit point), if every neighborhood of $r$ contains infinitely many distinct points of $\mathbf{S}$.

Basic Examples:
$(a, b),\ [a, b],\ (a, b],\ [a, b)$ have the same result, that is, the boundary is $\{a, b\}$, the interior is $(a, b)$, no isolated points, and the accumulation points are the set itself.
The boundary of the set $\mathbb{R}$ as well as its interior is the set $\mathbb{R}$ itself. No point is isolated, all points are accumulation points.
The boundary of the empty set $\varnothing$ as well as its interior is the empty set $\varnothing$ itself. Since the set contains no points, it can not contain isolated or accumulation points.