# Math Help - logic

1. ## logic

Use the 17 Rules of Inference to prove the following arguments valid.
Question 1.
1. (Q . F) v R
2. (Q v R) > ~P
3. ~A / ~(A v P)

Question 2.
1. A > ~A / ~A

Question 3.
1. A > B
2. C > B / (A v C) > B

Question 4.
1. S > R/ S > (R v T)

??? Ah

2. Hello, scoober!

Here's #3 . . .

We have the rule: . $(p \to q) \;\Longleftrightarrow \;(\sim p \vee q)$

I will call it "ADI" -- Alternate Definition of Implication.

Use the 17 Rules of Inference to prove the following argument valid.

$3.\;\;\begin{array}{c}A \to B \\ C \to B \\ \hline (A \vee C) \to B \end{array}$

. . $\begin{array}{ccccc} & \text{Statement} && \text{Reason} \\ \hline \\[-3mm]
1. & (A \to B) \wedge (C \to B) & & \text{Given} \\ \\[-3mm]
2. & (\sim A \vee B) \wedge (\sim C \vee B) & & \text{ADI} \\ \\[-3mm]
3. & (\sim A \:\wedge \sim C) \vee B & & \text{Distr.} \\ \\[-3mm]
4. & \sim(A \vee C) \vee B & & \text{DeMorgan} \\ \\[-3mm]
5. & (A \vee C) \to B && \text{ADI}\end{array}$