1.4 Logic & Propositions

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

     P AND NOT(Q) NOT(P AND Q) P OR NOT(Q) NOT(P) AND Q NOT(P) OR Q NOT(P) OR Q 
   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

     (P OR Q) AND (NOT(P) OR NOT(Q)) (P AND Q) OR (NOT(P) AND NOT(Q)) NOT(P AND Q) AND (NOT(P) OR NOT(Q)) NOT(P AND Q) OR NOT(P) OR NOT(Q) (P AND Q) OR (NOT(P) AND NOT(Q)) 
   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 IMPLIES NOT(Q)) AND (NOT(P) IMPLIES Q) (P AND NOT(Q)) OR (NOT(P) AND Q) (P OR NOT(Q)) AND (NOT(P) OR Q) (P AND Q) OR (NOT(P) AND NOT(Q)) NOT(P AND Q) (P AND NOT(Q)) OR (NOT(P) AND Q) 
   \((P \text{ IMPLIES } \text{NOT}(Q)) \text{ AND } (\text{NOT}(P) \text{ IMPLIES } Q)\) 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 AND NOT(Q) NOT(P AND Q) NOT(P) AND NOT(Q) (P OR Q) AND (NOT(P) OR Q) (P OR Q) AND (P OR NOT(Q)) NOT(P) AND NOT(Q) 
   \( (P \text{ NOR } Q)\) 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.
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