\( \def\naturals{\mathbb{N}} \def\integers{\mathbb{Z}} \def\rationals{\mathbb{Q}} \def\reals{\mathbb{R}} \def\complex{\mathbb{C}} \)
Topics: Reals, Complex field, Counting
Book Chapters: Rudin, Chapters 1 and 2
Due date: Before class, 23 Sep 2015
Mode of submissions: Please submit at blackboard.
Format: Please keep it in pdf. format with your name and andrew id included in it. The filename must be LastName_FirstName-yourAndrewID-HW2.
Expected time required for this homework: Six hours. If you're quick, you can do it in under an hour.

I. The Reals

  1. Are $\sqrt{3}$ and $\sqrt{5}$ rational numbers? And how about $\sqrt{5} - \sqrt{3}$? Justify your answer with proofs.
  2. Prove that
    1. between any two real numbers, there is a rational one,
    2. between any two real numbers, there are uncountable number of irrationals.

II. The Complex Field

  1. Let $a$ and $b$ be complex numbers ($a,b \in \complex$), prove that
    1. $|a| \ge 0$,
    2. $|\overline a| = |a|$,
    3. $|ab| = |a||b|$,
    4. $|\operatorname{Re}(a)| \le |a|$,
    5. $|a + b| \le |a| + |b|$.
  2. Let $Z_1, Z_2, Z_3,\dots,Z_n \in \complex$.
    1. Prove that \[ |Z_1 + Z_2 + Z_3 + \dots + Z_n| \le |Z_1|+ |Z_2| + |Z_3| + \dots + |Z_n|. \]
    2. Prove that \[ |\lt Z_1,Z_2 \gt + \lt Z_2,Z_3 \gt + \cdots + \lt Z_{n-1},Z_n \gt + \lt Z_n,Z_1 \gt|~\leq~|Z_1|^2 + |Z_2|^2 + \cdots + |Z_n|^2\]

III. Counting

  1. Is the set of all irrational real numbers countable? Prove your answer.
  2. Prove that every infinite subset of a countable set is countable.
  3. Show that any finite set of reals includes its infimum. You must use induction for this proof.
  4. Show that any finite set of reals includes its supremum.