Let's break down the truth value of the statement [tex]\( (p \wedge \sim q) \wedge r \)[/tex] step by step under the given conditions. Here, [tex]\(\wedge\)[/tex] denotes the logical AND operation, and [tex]\(\sim q\)[/tex] denotes the logical negation of [tex]\(q\)[/tex].
### Condition (a)
[tex]\( p \)[/tex] is false, [tex]\( q \)[/tex] is false, and [tex]\( r \)[/tex] is true.
1. Determine [tex]\(\sim q\)[/tex]:
- Since [tex]\( q \)[/tex] is false, [tex]\(\sim q\)[/tex] (not [tex]\( q \)[/tex]) is true.
2. Evaluate [tex]\( p \wedge \sim q \)[/tex]:
- [tex]\( p \)[/tex] is false.
- [tex]\(\sim q \)[/tex] is true.
- [tex]\( p \wedge \sim q \)[/tex] (false [tex]\(\wedge\)[/tex] true) is false, since AND operation is true only if both operands are true.
3. Finally, evaluate [tex]\( (p \wedge \sim q) \wedge r \)[/tex]:
- [tex]\( p \wedge \sim q \)[/tex] is false.
- [tex]\( r \)[/tex] is true.
- [tex]\((p \wedge \sim q) \wedge r \)[/tex] (false [tex]\(\wedge\)[/tex] true) is false, since AND operation is true only if both operands are true.
Hence, the truth value of this statement under condition (a) is false.
### Condition (b)
[tex]\( p \)[/tex] is true, [tex]\( q \)[/tex] is false, and [tex]\( r \)[/tex] is false.
1. Determine [tex]\(\sim q\)[/tex]:
- Since [tex]\( q \)[/tex] is false, [tex]\(\sim q\)[/tex] (not [tex]\( q \)[/tex]) is true.
2. Evaluate [tex]\( p \wedge \sim q \)[/tex]:
- [tex]\( p \)[/tex] is true.
- [tex]\(\sim q \)[/tex] is true.
- [tex]\( p \wedge \sim q \)[/tex] (true [tex]\(\wedge\)[/tex] true) is true.
3. Finally, evaluate [tex]\( (p \wedge \sim q) \wedge r \)[/tex]:
- [tex]\( p \wedge \sim q \)[/tex] is true.
- [tex]\( r \)[/tex] is false.
- [tex]\((p \wedge \sim q) \wedge r \)[/tex] (true [tex]\(\wedge\)[/tex] false) is false, since AND operation is true only if both operands are true.
Hence, the truth value of this statement under condition (b) is false.
### Answer
Under both given conditions, the truth value of the statement [tex]\( (p \wedge \sim q) \wedge r \)[/tex] is false.
So for condition (a):
If [tex]\( p \)[/tex] is false, [tex]\( q \)[/tex] is false, and [tex]\( r \)[/tex] is true, the truth value of [tex]\( (p \wedge \sim q) \wedge r \)[/tex] is false.
