Problem 1 Given the sets
$A = \{a, b, c, d\}$, $B = \{b, c, d, e, f\}$, $C = \{c, e, g, h\}$,
determine which of the following are true:
1. $b \in A \cap B$
2. $(A \cup B) \cap C = \{c, e\}$
3. $(A \setminus B) \cap C = \{a\}$
4. $A \cap C \subseteq B$
Problem 2 Prove by induction that for all integers $n \ge 5$, $2^n > n^2$.
Problem 3 Prove by contradiction that $\sqrt{3}$ is irrational.
Problem 4 Simplify:
$\frac{1}{x + \sqrt{x}} - \frac{1}{\sqrt{x}}$
Problem 5 If
$(1 + \frac{2}{n})(1 - \frac{1}{m}) = 1$,
find $m$ in terms of $n$.
Problem 6 Evaluate
$\sum_{k=1}^{n} (5 + 3k)$.
Problem 7 Consider the macro model
(i) $Y = C + \bar{I} + G$, (ii) $C = a + b(Y - T) + \alpha Y$, (iii) $T = d + tY$,
where $b, t \in (0, 1)$ and $a, \bar{I}, d, G, \alpha$ are parameters.
1. Express $Y$ in terms of $\bar{I}, G$, and the parameters.
Problem 8 For what values of $x$ is $|x^2 - 5| \ge 4$?
Problem 9 Prove by induction that for all integers $n \ge 1$,
$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.
