Answer :
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].
