Get the answers you've been looking for with the help of IDNLearn.com's expert community. Our platform is designed to provide trustworthy and thorough answers to any questions you may have.
Sagot :
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
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.