Question

Complete the formal proof of (~Q→~R)v(R&~Q) from no premises. The empty premise line is not numbered. Hint: there are longer ways of doing the proof that require the 5-step plan in the middle somewhere, but we require to you find the shortcuts and do it in 11 lines.

Use -> for arrow, # for contradiction; justify subproof assumptions with Assume; always drop outer parentheses; no spaces in PROP.

Answer 1

Answer:

Step-by-step explanation:

Try something like this but in the notation that you're using.

1. | ~( (~Q ->~R) v (R & ~Q) ) Assume

2. | |  ~(~Q ->~R) Assume

3. | | |  ~(R & ~Q) Assume

4. | | |  ~R v Q    3, De Morgan

5. | | |  ~Q -> ~R 4 Material implication

6. | | |  # 2, 5 Contradiction

7. | | R & ~Q 3-6 indirect Proof

8. | ~(~Q ->~R) ->   (R & ~Q ) 2-7 Cond. Proof

9. | (~Q ->~R) v (R & ~Q) 8 Material implication

10 | # 1,9 Contradiction    

11.  (~Q ->~R) v (R & ~Q) 1-10 Indirect proof

Answer 2

As per the question the formal proof of the (~Q→~R)v(R&~Q) is having no premise and is an empty premise line is not a numbered. Hence the longer the ways of doing the proof need a five step plan middle somewhere

• In first ~( (~Q ->~R) v (R & ~Q) ) Assume  |  ~(~Q ->~R) Assume . Next  | | |  ~(R & ~Q) Assume  then | | |  ~R v Q    3, De Morgan . Here | | |  ~Q -> ~R 4 Material implication  and | | |  # 2, 5 Contradiction

• Hence the (~Q ->~R) v (R & ~Q) 1-10 thus a Indirect proof.

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