Supply steps in a proof of a valid argument and give reasons using rules of inference.
1.)~A*B
2.)C>A
3.)CvD/D
To prove the validity of the argument, we will use the rules of inference. The objective is to derive the conclusion (D) from the premises (1, 2, and 3). Here are the steps for the proof:
1. ~A * B (Premise)
2. C > A (Premise)
3. C v D (Premise)
To begin, we will consider the third premise, which states that either C or D is true. Since we do not know which one is true, we will consider both cases separately to see if either one leads us to the conclusion.
Case 1: C is true
4. C (Assumption)
5. A (Modus Ponens: 2, 4)
6. ~A (Simplification: 1)
7. Contradiction: A and ~A (5, 6)
8. D (Ex Falso Quodlibet: 7)
9. D (Disjunction Introduction: 8)
At this point, we have derived the conclusion (D) from the first case (C is true). Now, we will consider the second case.
Case 2: D is true
10. D (Assumption)
11. D (Reiteration: 10)
In this case, we have directly obtained the conclusion (D) from the second premise, which makes it trivial.
Since we have derived the conclusion (D) in both cases, regardless of whether C or D is true, we can conclude that the argument is valid.