Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Let's interpret the question step by step and fill in the truth values for the logical expression [tex]\((p \vee \sim q) \wedge r\)[/tex].
### Given Values:
1. [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex] are the well-known truth values for logical propositions.
2. We need to compute the truth values for the expression [tex]\((p \vee \sim q) \wedge r\)[/tex] for each combination of [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex].
### Logical Components:
- [tex]\(\vee\)[/tex]: Logical OR operator. [tex]\(A \vee B\)[/tex] is true if either [tex]\(A\)[/tex] or [tex]\(B\)[/tex] is true.
- [tex]\(\sim q\)[/tex]: Logical NOT operator. [tex]\(\sim q\)[/tex] is true if [tex]\(q\)[/tex] is false, and vice versa.
- [tex]\(\wedge\)[/tex]: Logical AND operator. [tex]\(A \wedge B\)[/tex] is true if both [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are true.
### Combinations of [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex]:
By the structure of the truth table given, let's consider all pairs [tex]\((p, q, r)\)[/tex] and analyze the expression [tex]\((p \vee \sim q) \wedge r\)[/tex].
### Step-by-Step Analysis:
#### [tex]\(T, T, T\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(F\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(T \vee F = T\)[/tex]
3. [tex]\((T \vee F) \wedge T\)[/tex] ⇒ [tex]\(T \wedge T = T\)[/tex]
Result: [tex]\(T\)[/tex]
#### [tex]\(T, T, F\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(F\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(T \vee F = T\)[/tex]
3. [tex]\((T \vee F) \wedge F\)[/tex] ⇒ [tex]\(T \wedge F = F\)[/tex]
Result: [tex]\(F\)[/tex]
#### [tex]\(T, F, T\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(T\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(T \vee T = T\)[/tex]
3. [tex]\((T \vee T) \wedge T\)[/tex] ⇒ [tex]\(T \wedge T = T\)[/tex]
Result: [tex]\(T\)[/tex]
#### [tex]\(T, F, F\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(T\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(T \vee T = T\)[/tex]
3. [tex]\((T \vee T) \wedge F\)[/tex] ⇒ [tex]\(T \wedge F = F\)[/tex]
Result: [tex]\(F\)[/tex]
#### [tex]\(F, T, T\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(F\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(F \vee F = F\)[/tex]
3. [tex]\((F \vee F) \wedge T\)[/tex] ⇒ [tex]\(F \wedge T = F\)[/tex]
Result: [tex]\(F\)[/tex]
#### [tex]\(F, T, F\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(F\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(F \vee F = F\)[/tex]
3. [tex]\((F \vee F) \wedge F\)[/tex] ⇒ [tex]\(F \wedge F = F\)[/tex]
Result: [tex]\(F\)[/tex]
#### [tex]\(F, F, T\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(T\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(F \vee T = T\)[/tex]
3. [tex]\((F \vee T) \wedge T\)[/tex] ⇒ [tex]\(T \wedge T = T\)[/tex]
Result: [tex]\(T\)[/tex]
#### [tex]\(F, F, F\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(T\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(F \vee T = T\)[/tex]
3. [tex]\((F \vee T) \wedge F\)[/tex] ⇒ [tex]\(T \wedge F = F\)[/tex]
Result: [tex]\(F\)[/tex]
### Completed Truth Table:
[tex]\[ \begin{array}{|c|c|c|c|} \hline p & q & r & (p \vee \sim q) \wedge r \\ \hline T & T & T & T \\ T & T & F & F \\ T & F & T & T \\ T & F & F & F \\ F & T & T & F \\ F & T & F & F \\ F & F & T & T \\ F & F & F & F \\ \hline \end{array} \][/tex]
So the correct sequence of truth values for the given expression is [tex]\([T, F, T, F, F, F, T, F]\)[/tex].
### Given Values:
1. [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex] are the well-known truth values for logical propositions.
2. We need to compute the truth values for the expression [tex]\((p \vee \sim q) \wedge r\)[/tex] for each combination of [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex].
### Logical Components:
- [tex]\(\vee\)[/tex]: Logical OR operator. [tex]\(A \vee B\)[/tex] is true if either [tex]\(A\)[/tex] or [tex]\(B\)[/tex] is true.
- [tex]\(\sim q\)[/tex]: Logical NOT operator. [tex]\(\sim q\)[/tex] is true if [tex]\(q\)[/tex] is false, and vice versa.
- [tex]\(\wedge\)[/tex]: Logical AND operator. [tex]\(A \wedge B\)[/tex] is true if both [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are true.
### Combinations of [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex]:
By the structure of the truth table given, let's consider all pairs [tex]\((p, q, r)\)[/tex] and analyze the expression [tex]\((p \vee \sim q) \wedge r\)[/tex].
### Step-by-Step Analysis:
#### [tex]\(T, T, T\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(F\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(T \vee F = T\)[/tex]
3. [tex]\((T \vee F) \wedge T\)[/tex] ⇒ [tex]\(T \wedge T = T\)[/tex]
Result: [tex]\(T\)[/tex]
#### [tex]\(T, T, F\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(F\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(T \vee F = T\)[/tex]
3. [tex]\((T \vee F) \wedge F\)[/tex] ⇒ [tex]\(T \wedge F = F\)[/tex]
Result: [tex]\(F\)[/tex]
#### [tex]\(T, F, T\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(T\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(T \vee T = T\)[/tex]
3. [tex]\((T \vee T) \wedge T\)[/tex] ⇒ [tex]\(T \wedge T = T\)[/tex]
Result: [tex]\(T\)[/tex]
#### [tex]\(T, F, F\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(T\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(T \vee T = T\)[/tex]
3. [tex]\((T \vee T) \wedge F\)[/tex] ⇒ [tex]\(T \wedge F = F\)[/tex]
Result: [tex]\(F\)[/tex]
#### [tex]\(F, T, T\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(F\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(F \vee F = F\)[/tex]
3. [tex]\((F \vee F) \wedge T\)[/tex] ⇒ [tex]\(F \wedge T = F\)[/tex]
Result: [tex]\(F\)[/tex]
#### [tex]\(F, T, F\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(F\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(F \vee F = F\)[/tex]
3. [tex]\((F \vee F) \wedge F\)[/tex] ⇒ [tex]\(F \wedge F = F\)[/tex]
Result: [tex]\(F\)[/tex]
#### [tex]\(F, F, T\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(T\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(F \vee T = T\)[/tex]
3. [tex]\((F \vee T) \wedge T\)[/tex] ⇒ [tex]\(T \wedge T = T\)[/tex]
Result: [tex]\(T\)[/tex]
#### [tex]\(F, F, F\)[/tex]
1. [tex]\(\sim q\)[/tex] ⇒ [tex]\(T\)[/tex]
2. [tex]\(p \vee \sim q\)[/tex] ⇒ [tex]\(F \vee T = T\)[/tex]
3. [tex]\((F \vee T) \wedge F\)[/tex] ⇒ [tex]\(T \wedge F = F\)[/tex]
Result: [tex]\(F\)[/tex]
### Completed Truth Table:
[tex]\[ \begin{array}{|c|c|c|c|} \hline p & q & r & (p \vee \sim q) \wedge r \\ \hline T & T & T & T \\ T & T & F & F \\ T & F & T & T \\ T & F & F & F \\ F & T & T & F \\ F & T & F & F \\ F & F & T & T \\ F & F & F & F \\ \hline \end{array} \][/tex]
So the correct sequence of truth values for the given expression is [tex]\([T, F, T, F, F, F, T, F]\)[/tex].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
