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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
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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