**3.**Valid? Prove or disprove.

**4.**Valid? Prove or disprove.

If I work hard, then I earn lots of money.

**5.**True or False questions.

(i) If $p ∧ q$ is true, then $p ∨ q$ is true.

(ii) If $p → q$ is true and $q \to p$ is true, then $p$ is logically equivalent to $q$.

(iii) If $A$ is a tautology and $B$ is a contradiction, then $A ∧ (¬B)$ is a tautology.

(iv) If $A \iff B$ and $C$ is any statement, then $(A\to C) \iff (B \to C)$.

(v) If the premises of an argument are all contradictions, then the argument is valid.

(vi) The statement $(p \to q) \iff (q ∧ (r \to s))$ evaluates to true when all the atomic statements $p, q, r, s$ are true.

**6.**In the math department there are 30 personal computers (PCs). $\begin{align} 20 \quad & \text{have A drives} \\ 8 \quad & \text{ have 19-inch monitors} \\ 25 \quad & \text{are running Windows XP} \\ 20 \quad & \text{have at least two of these properties} \\ 6 \quad & \text{have all three properties} \end{align}$

**7.**How many ways can you get a total of 6 when rolling two dice?

**8.**How many three digit numbers contain the digits $2$ and $5$ but not $0, 3$, or $7$?

**9.**In a group of 29 people, how many people must there be whose birthdays are in the same month?

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