# 1.4 Logic & Propositions

## Logical Connectives

For each of the following formulas, choose the $$\textrm{AND}$$, $$\textrm{OR}$$, and $$\textrm{NOT}$$ formula equivalent to it.

1. $$P \textrm{ IMPLIES } Q$$

Exercise 1
An implication is true exactly when the if-part, in this case $$P$$, is FALSE, or the then-part, in this case $$Q$$, is TRUE.
2. $$P \textrm{ IFF } Q$$

Exercise 2
If $$P$$ is TRUE, the answer formula simplifies to $$Q$$, so the formula is TRUE only if $$Q$$ is also TRUE, that is, only if $$Q$$ has the same truth value as $$P$$. Likewise, if $$P$$ is FALSE, the answer simplifies to $$\text{NOT}(Q)$$, so again it is TRUE only if $$Q$$ has the same truth value of $$P$$, namely FALSE.
3. $$P \text{ XOR } Q$$

Exercise 3
$$(P \text{ IMPLIES } \text{NOT}(Q)) \text{ AND } (\text{NOT}(P) \text{ IMPLIES } Q)$$ is equivalent to $$P \text{ XOR } Q$$, but it is not an $$\text{AND}$$, $$\text{OR}$$, and $$\text{NOT}$$ formula.
4. $$P \text{ NOR } Q$$

Exercise 4
$$(P \text{ NOR } Q)$$ is $$\text{NOT}(P \text{ OR } Q)$$. The truth table for $$\text{NOR}$$ will produce a TRUE output only when both $$P$$ and $$Q$$ are FALSE. Thus, $$\text{NOT}(P) \text{ AND } \text{NOT}(Q)$$ is a correct answer.