Determine whether the following are correct or erroneous application of our disj

Question

Determine whether the following are correct or erroneous application of our disjunction rules. State what the derivations are for each one.

(1)
1. P v (Q ->R)               premise
2. ~Q -> R                     premise
3. P                                1,2 DS

(2)
1. P v (Q->R)                 premise
2. ~(Q->R)                     premise
3. P                                 1,2 DS

(3)

1. P                                  premise
2. (P v Q) v (R v S)         1 vl

(4)
1. P                                  premise
2. P v (Q v (R v S)           1 vl

(5)
1. (P & R) v Q                 premise
2. (P & R) -> (A v B)       premise
3. Q -> (A v B)                 premise
4. A v B                             1,2,3 vE

(6)
1. (P & R) v Q                  premise
2. (P & R) -> A                 premise
3. Q-> B                            premise
4. A v B                             1,2,3 vE

Explanation / Answer

(1)

=> P v (Q ->R)

=>  ~Q->R

these are the 2 premises given

we need to deduce P from these 2

As ~Q->R   

=>  P v (Q ->~R)

=> P v (~R)

=> P

(2)
=> P v (Q->R)                 premise
=> ~(Q->R)                     premise

=> so ~Q implies ~R

if we think Q->R as true then ~(Q->R) is false

then P v (False) => P

if we think Q->R as False then ~(Q->R) is True

then P v (True) => True

(3)

1. P                                  premise
2. (P v Q) v (R v S)         1 vl

T

(4)
1. P                                  premise
2. P v (Q v (R v S)           1 vl

T

P Q R S (PvQ) (RvS) (PvQ)v(RvS) T T T T T T T T T T F T T T T T F T T T T T T F F T F T

T

F T T T T T T F T F T T T T F F T T T T T F F F T F T F T T T T T T F T T F T T T F T F T T T T F T F F T F T F F T T F T T F F T F F T T F F F T F T T F F F F F F F

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