Appendix A — Mathematical Foundations

A.1 Sets

The following statements are True or False?

Exercise A.1 \mathbb{Q}\subsetneq\mathbb{R}?

Exercise A.2 \mathbb{Z}\subset\mathbb{N}

Exercise A.3 A\cap\emptyset=\emptyset

Exercise A.4 A\cup B\subsetneq B

Exercise A.5 For two finite sets A and B, we have |A\cup B|=|A|+|B|.

Exercise A.6 A\cap(B\cup C)=(A\cap B)\cup(A\cap C). Draw to illustrate.

Exercise A.7 (A\cup B)^c=A^c\cap B^c.

Exercise A.8 For a finite set A, its power set, denoted by \mathcal{P}(A), has 2^{|A|} many elements.

Exercise A.9 Draw the graphs of \log_{e}(x), the natural logarithm function.

Exercise A.10 How is the above graph different from the graph of \log_{2}(x)?

Exercise A.11 Draw the graph of x^2, \sqrt{x}, and x in the same plot.

Exercise A.12 Can you find an m such that the line y=mx stays above the graph of \sqrt{x} for any large positive x?

Exercise A.13 Can you find an m such that the line y=mx stays above the graph of x^2 for any large positive x?

Sum of natural numbers: \sum_{k=1}^n k=\frac{n(n+1)}{2}. \tag{A.1}
Finite geometric series: \sum_{k=0}^n x^k=\frac{x^{n+1}-1}{x-1}. \tag{A.2}
Infinite geometric series for |x|<1: \sum_{k=0}^\infty x^k=\frac{1}{1-x}. \tag{A.3}
For |x|<1: \sum_{k=0}^\infty kx^k=\frac{x}{(1-x)^2}. \tag{A.4}

Exercise A.14 What is mathematical induction?

Exercise A.15 Prove the first identity above using mathematical induction.