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The truth table represents statements [tex]p, q[/tex], and [tex]r[/tex]. Which rows represent when [tex](p \wedge q) \vee (p \wedge r)[/tex] is true?

\begin{tabular}{|c|c|c|c|c|c|}
\hline & [tex]$p$[/tex] & [tex]$q$[/tex] & [tex]$r$[/tex] & [tex]$p \wedge q$[/tex] & [tex]$p \wedge r$[/tex] \\
\hline A & T & T & T & T & T \\
\hline B & T & T & F & T & F \\
\hline C & T & F & T & F & T \\
\hline D & T & F & F & F & F \\
\hline E & F & T & T & F & F \\
\hline F & F & T & F & F & F \\
\hline G & F & F & T & F & F \\
\hline H & F & F & F & F & F \\
\hline
\end{tabular}

A. A and B

B. A, B, and C

C. B and E

D. B, C, and E



Answer :

GinnyAnswer
To determine which rows represent when [tex]\((p \wedge q) \vee (p \wedge r)\)[/tex] is true, we need to analyze the truth table.

Let's break down what [tex]\((p \wedge q) \vee (p \wedge r)\)[/tex] means:
- [tex]\((p \wedge q) \)[/tex] is true when both [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are true.
- [tex]\((p \wedge r)\)[/tex] is true when both [tex]\(p\)[/tex] and [tex]\(r\)[/tex] are true.
- [tex]\((p \wedge q) \vee (p \wedge r)\)[/tex] is true when either [tex]\((p \wedge q)\)[/tex] is true, or [tex]\((p \wedge r)\)[/tex] is true, or both are true.

Now we will check each row to see if [tex]\((p \wedge q)\)[/tex] or [tex]\((p \wedge r)\)[/tex] is true:

1. Row A: [tex]\(p = T\)[/tex], [tex]\(q = T\)[/tex], [tex]\(r = T\)[/tex], [tex]\(p \wedge q = T\)[/tex], [tex]\(p \wedge r = T\)[/tex]
- [tex]\((p \wedge q) \vee (p \wedge r) = T \vee T = T\)[/tex]
- So, Row A is included.

2. Row B: [tex]\(p = T\)[/tex], [tex]\(q = T\)[/tex], [tex]\(r = F\)[/tex], [tex]\(p \wedge q = T\)[/tex], [tex]\(p \wedge r = F\)[/tex]
- [tex]\((p \wedge q) \vee (p \wedge r) = T \vee F = T\)[/tex]
- So, Row B is included.

3. Row C: [tex]\(p = T\)[/tex], [tex]\(q = F\)[/tex], [tex]\(r = T\)[/tex], [tex]\(p \wedge q = F\)[/tex], [tex]\(p \wedge r = T\)[/tex]
- [tex]\((p \wedge q) \vee (p \wedge r) = F \vee T = T\)[/tex]
- So, Row C is included.

4. Row D: [tex]\(p = T\)[/tex], [tex]\(q = F\)[/tex], [tex]\(r = F\)[/tex], [tex]\(p \wedge q = F\)[/tex], [tex]\(p \wedge r = F\)[/tex]
- [tex]\((p \wedge q) \vee (p \wedge r) = F \vee F = F\)[/tex]
- So, Row D is not included.

5. Row E: [tex]\(p = F\)[/tex], [tex]\(q = T\)[/tex], [tex]\(r = T\)[/tex], [tex]\(p \wedge q = F\)[/tex], [tex]\(p \wedge r = F\)[/tex]
- [tex]\((p \wedge q) \vee (p \wedge r) = F \vee F = F\)[/tex]
- So, Row E is not included.

6. Row F: [tex]\(p = F\)[/tex], [tex]\(q = T\)[/tex], [tex]\(r = F\)[/tex], [tex]\(p \wedge q = F\)[/tex], [tex]\(p \wedge r = F\)[/tex]
- [tex]\((p \wedge q) \vee (p \wedge r) = F \vee F = F\)[/tex]
- So, Row F is not included.

7. Row G: [tex]\(p = F\)[/tex], [tex]\(q = F\)[/tex], [tex]\(r = T\)[/tex], [tex]\(p \wedge q = F\)[/tex], [tex]\(p \wedge r = F\)[/tex]
- [tex]\((p \wedge q) \vee (p \wedge r) = F \vee F = F\)[/tex]
- So, Row G is not included.

8. Row H: [tex]\(p = F\)[/tex], [tex]\(q = F\)[/tex], [tex]\(r = F\)[/tex], [tex]\(p \wedge q = F\)[/tex], [tex]\(p \wedge r = F\)[/tex]
- [tex]\((p \wedge q) \vee (p \wedge r) = F \vee F = F\)[/tex]
- So, Row H is not included.

After our detailed analysis, the rows where [tex]\((p \wedge q) \vee (p \wedge r)\)[/tex] is true are:
- A
- B
- C

Therefore, the answer is:
A, B, and C

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