SOLUTION: Prove using Indirect Proof 1. B ⊃ (K • M) 2. (B • M) ⊃ (P ≡ ∼P) / ∼B

Algebra.Com
Question 1193814: Prove using Indirect Proof
1. B ⊃ (K • M)
2. (B • M) ⊃ (P ≡ ∼P) / ∼B
Found 3 solutions by Alan3354, math_tutor2020, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
10 similar problems posted in succession.
Maybe you're in the wrong class?
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

This is one way to do the derivation using an indirect proof (aka proof by contradiction).
NumberStatementLine(s) UsedReason
1B -> (K & M)
2(B & M) -> (P = ~P)
:.~B
3BAssumption for Indirect Proof
4K & M1,3Modus Ponens
5M4Simplification
6B & M3, 5Conjunction
7P = ~P2, 6Modus Ponens
8(P -> ~P) & (~P -> P)7Material Equivalence
9P -> ~P8Simplification
10~P v ~P9Material Implication
11~P10Taulogy
12~P -> P8Simplification
13~~P v P12Material Implication
14P v P 13Double Negation
15P14Taulogy
16P & ~P15, 11Conjunction
17~B3 - 16Indirect Proof

The conclusion we want to arrive at is ~B
Assume that the opposite is the case and we assume B (line 3)
The idea is to see if it generates a contradiction.

The line P & ~P is a contradiction because one portion is true while the other is false, which makes P & ~P always false.
Or perhaps another approach is to look at line 7 where we have P = ~P. This isn't possible because if P is true then ~P is false, and vice versa. There's no way to have true equal false. So we could have stopped at that point to find the contradiction.

Therefore, the negation of the assumption must be the case and we have ~B instead as the proper conclusion.

I used arrows in place of horseshoe symbols.
I used ampersands (&) in place of the dots.
Instead of a triple equal sign, I used a regular equal sign.

Answer by ikleyn(52777)   (Show Source): You can put this solution on YOUR website!
.

When a person comes with 5 or 10 same type problems, it is clear that learning is not his (or her) goal.



RELATED QUESTIONS

Complete the following proofs using the Conditional Proof method. 1) 1. ~(P ∙... (answered by solver91311)
Use conditional proof: 1. S ⊃ (B ⊃ T) 2. N ⊃ (T ⊃ ∼B) / (S • N) ⊃... (answered by jim_thompson5910)
Prove the following three arguments to be valid using the method of Natural Deduction A. (answered by solver91311)
Use indirect proof: 1. (P ∨ F) ⊃ (A ∨ D) 2. A ⊃ (M • ∼P) 3. D ⊃ (C (answered by Edwin McCravy)
Prove the following using indirect proof: 1. (A ∨ B) ⊃ C 2. (∼A ∨ D) ⊃ E (answered by math_tutor2020)
COMPLETE THE FOLLOWING PROOFS WITHOUT USING CONDITIONAL PROOF OR INDIRECT PROOF (10 PTS. (answered by Edwin McCravy)
1. H > (M > E) 2. ~[(M > H) > E] 3. (I + O) > [M + (X = P)] 4. [(K v B) > ~M] > (~I (answered by CPhill)
1. (J v F) v M 2. (J v M) ⊃ ~ P 3. ~F/~(F v P) 4. M Assumption... (answered by math_tutor2020)
Use an ordinary proof (not conditional or indirect) to solve the following arguments... (answered by Edwin McCravy)