b. Let p : I have a headache. Let \( \mathrm{q}: \) I drink enough water. Let r : I need to take medicine. Given the statement "If I have a headache, then I did not drink enough water and I need to take medicine." The conditional statement is \[ \sim p \rightarrow(\sim q \wedge r) \] \( \sim \sim p \rightarrow \sim(q \wedge r) \) \( p \rightarrow(\sim q \wedge r) \) p \( \rightarrow \sim(\sim q \wedge \sim r) \)

The statement "If I have a headache, then I did not drink enough water and I need to take medicine" can be represented as \( p \rightarrow (\sim q \wedge r) \). This means that if the condition \( p \) (having a headache) is true, then both \( \sim q \) (not drinking enough water) and \( r \) (needing to take medicine) must also be true. It's like a domino effect—once you hit the headache, you better believe the water-drinking and medicine-taking are in play! When dealing with logical statements like these, a common mistake is to misinterpret the negation of the statements. For example, confusing \( \sim(q \wedge r) \) with \( \sim q \wedge \sim r \) can lead directly to flawed conclusions. Remember, De Morgan's laws tell us that \( \sim(q \wedge r) \) becomes \( \sim q \vee \sim r \)—so if either part is false, the whole combination can hold true. Logic can be tricky, but once you grasp these concepts, you'll be a reasoning superstar!