Complete the formal proof of (~Q→~R)v(R&~Q) from no premises. The empty premise line is not numbered. Hint: there are longer

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2 months ago

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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.

Mathematics

1 answer:

tester [12.3K] · Answer 1 · 2026-03-29T00:26:43+03:00

Answer:

Step-by-step breakdown:

You may consider a structure like the following while keeping in mind your specific notation.

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 Results in a contradiction

7. | | R & ~Q 3-6 indirect proof outcome

8. | ~(~Q ->~R) -> (R & ~Q ) 2-7 Discharging the conditional proof

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

10 | # 1,9 Results in contradiction

11. (~Q ->~R) v (R & ~Q) 1-10 Completion via indirect proof