Unit 4 Bounds and Completeness [Lecture PDF]
1. Which one of the following statements is TRUE?
A. It is possible that a set has only one upper bound
B. It is possible that a bounded set does not have a greatest lower bound
C. If a set has an upper bound, then it must have a least upper bound
D. If a set has an upper bound, then it is bounded
2. Given the set of numbers $S=\{1, 1.1, 0.9, 1.01, 0.99, 1.001, 0.999, \ldots\}$
A. What is the least upper bound of $S$?
B. What is the greatest lower bound of $S$?
C. What is a limit point of $S$?
Solutions:
1. C is correct.
A. If a set has an upper (lower) bound then it must have infinite upper (lower) bounds.
B. If a set is bounded, then it must have both upper bounds and lower bounds, and therefore the supremum (least upper bound) and infimum (greatest lower bound).
D. Only if a set has both upper bounds and lower bounds, the set is bounded.
2.
A. The least upper boung of $S$: $\sup S=1.1$
B. The greatest lower bound of $S$: $\inf S=0.9$
C. The limit point of $S$: $\lim S=1$
Unit 5 Complex Number System, Algebraic and Transcendental Numbers [Lecture PDF]
1. Which of the following statements are TRUE?
A. A polynomial equation of degree $n$ must have, counted with multiplicity, exactly $n$ roots in $\mathbb{R}$, where $n$ is a positive integer
B. Every rational number is algebraic
C. Every irrational number is transcendental
D. $e$ is irrational as well as transcendental
E. Any number in $\mathbb{C}$ cannot be a real number
2. Given following numbers: $$e, \pi, 0, \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}, \sqrt{2}+\sqrt{3}+\sqrt{5}, 2+3i, \frac{4}{7}$$ Of all the numbers above,
A. What is the total number of algebraics?
B. What is the total number of transcendentals?
C. What is the total number of irrational numbers?
3. If $$\frac{25}{3-4i}+\frac{50}{4+3i}=a+bi$$ Then $a=?\ b=?$
Solutions:
1. B, D are correct.
A. A polynomial equation of degree $n$ must have, counted with multiplicity, exactly $n$ roots in $\mathbb{C}$, where $n$ is a positive integer.
C. Not all of irrational number is transcendental such as $\sqrt{2}$.
E. $\mathbb{R}\subset\mathbb{C}$.
2.
A. There are 5 algebraics: $$0, \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}, \sqrt{2}+\sqrt{3}+\sqrt{5}, 2+3i, \frac{4}{7}$$
B. There are 2 transcendental: $$e, \pi$$
C. There are 4 irrational numbers: $$e, \pi, \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}, \sqrt{2}+\sqrt{3}+\sqrt{5}$$
3. $$\frac{25}{3-4i}+\frac{50}{4+3i}=a+bi$$ $$\Rightarrow 3+4i+2\cdot(4-3i)=a+bi$$ $$\Rightarrow 11-2i=a+bi$$ $$\Rightarrow a=11, b=-2$$
QUIZ 1 (Chapter-1)
1. Consider the infinite set $S$ of numbers $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$ Which of the following statements about $S$ are TRUE?
A. $S$ is unbounded
B. -1 is a lower bound of $S$
C. 1 is the least upper bound of $S$
D. 0 is not the greatest lower bound of $S$
E. $S$ has no limit point in it
F. $S$ is a closed set
Solution:
B, C, E are correct.
A, D: $\sup S=1,\ \inf S=0$.
F: The limit point of $S$ is 0 yet $0\notin S$, hence it is not closed.
2. If $$(-1+\sqrt3i)^{10}=a+b\sqrt3i$$ then $a=?\ b=?$
Solution:
$$(-1+\sqrt3i)^{10}=(2e^{2\pi/3})^{10}$$ $$=2^{10}(\cos 20\pi/3+i \sin 20\pi/3)$$ $$=1024(\cos 2\pi/3+i \sin 2\pi/3)$$ $$=-512+512\sqrt3i$$ $$\Rightarrow a=-512, b=512$$
3. Calculate: $$|\sqrt2\frac{i+i^2+i^3+i^4+i^5}{1+i}|$$
Solution:
$$|\sqrt2\frac{i+i^2+i^3+i^4+i^5}{1+i}|$$ $$=|\sqrt2\cdot \frac{i\cdot(1+i+i^2+i^3+i^4)\cdot(i-1)}{(1+i)\cdot(i-1)}|$$ $$=|-\frac{\sqrt2}{2}\cdot i\cdot(i^5-1)|$$ $$=-\frac{\sqrt2}{2}\cdot(-1-i)=|\frac{\sqrt2}{2}+\frac{\sqrt2}{2}\cdot i|=1$$
4. Which one of the following equation is TRUE?
A. $3+3i=3\sqrt2 e^{\pi/4}$
B. $-1+\sqrt3i=2 e^{-2\pi i/3}$
C. $-1=e^{\pi i}$
D. $-2-2\sqrt3i=4 e^{2\pi i/3}$
Solution:
C is correct.
A: $3+3i=3\sqrt2 e^{\pi i/4}$
B: $-1+\sqrt3i=2 e^{2\pi i/3}$
D. $-2-2\sqrt3i=4 e^{4\pi i/3}$
5. Which one of the following statements is TRUE?
A. $\sqrt[3]{2}+\sqrt3$ is not an algebraic number
B. $\sqrt[3]{5}-\sqrt[4]{3}$ is not an irrational number
C. $\sqrt2+\sqrt3i$ is an irrational number
D. $a+bi$ may be a rational number
Solution:
D is correct.
A: Let $x=\sqrt[3]{2}+\sqrt3\Rightarrow (x-\sqrt3)^3=2\Rightarrow x^6-9x^4-4x^3+27x^2+36x-23=0$
B: $\sqrt[3]{5}-\sqrt[4]{3}$ is an irrational number.
C: $\sqrt2+\sqrt3i$ is not a real number at all so it cannot be an irrational number.
6. $a, b, c, d$ are real numbers. Which one of the following statements is NOT TRUE?
A. $a^2+b^2+c^2\geq ab+bc+ac$
B. If $a^2+b^2=1$, and $c^2+d^2=1$, then $ac+bd\leq 1$.
C. If $a>0$, then $a^{n+1}+\frac1{a^{n+1}}>a^n+\frac1{a^n}$, when $n$ is any positive integer.
D. If $a>c>0,b>d>0$, then $a+bi>c+di$.
Solution:
D is correct. Because there is no concept of inequality among complex numbers.
A: $$(a-b)^2+(b-c)^2+(c-a)^2\ge0$$ $$\Leftrightarrow 2\cdot(a^2+b^2+c^2)\ge2\cdot(ab+bc+ca)$$ $$\Leftrightarrow a^2+b^2+c^2\geq ab+bc+ac$$
B: $$(a-c)^2+(b-d)^2\ge0$$ $$\Leftrightarrow a^2+b^2+c^2+d^2-2\cdot(ac+bd)\ge0$$ $$\Leftrightarrow ac+bd\le1$$
C: $$a^{n+1}-a^n+\frac{1}{a^{n+1}}-\frac{1}{a^n}$$ $$=\frac{a^{2n+2}-a^{2n+1}+1-a}{a^{n+1}}$$ $$=\frac{(a^{2n+1}-1)(a-1)}{a^{n+1}}$$ the numerator is larger than 0 either $a>1$ or $0 < a < 1$.
7. Calculate: $$3\log_{\frac{1}{8}}{\frac1{128}}$$
Solution:
$$3\log_{\frac{1}{8}}{\frac1{128}}=3\log_{2^{-3}}{2^{-7}}$$ $$=3\cdot\frac{\lg{2^{-7}}}{\lg{2^{-3}}}=7$$
8. Which one of the following statements is NOT TRUE?
A. If $A=\{a_1,a_2,a_3,...\}$ is bounded, and $|a_n-a_{n+1}|\geq 1, a_n\neq a_m(n,m=1,2,3,...)$, then $A$ has at least one limit point.
B. Let $B$ be the set of all rational numbers in $(0,1)$ having denominator $2^n,n=1,2,3,...$, then $B$ is closed.
C. If $C$ is a closed set, $D$ is also a closed set, then $C\bigcup D$ is a closed set.
D. If $C$ is an open set, $D$ is also an open set, then $C\bigcup D$ is an open set.
Solution:
B is correct. Because 0 is a limit point of $B$, but 0 does not belong to $B$. Thus $B$ is not closed.
According to Bolzano-Weierstrass theorem, every bounded infinite set has at least one limit point.
[Reference Link1]
[Reference Link2]
9. Consider the interval $S=(-\infty, 1]$. Which one of the following statements about $S$ is NOT TRUE?
A. $S$ is a half open half closed set
B. Every point of $S$ is a limit point
C. $x=1$ is not an interior point of $S$
D. $S$ is bounded above, but is not bounded below.
Solution:
A is correct. There is no concept of "half open half closed" relating to a set.
C: $x=1$ is a boundary point of $S$.
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