I need help constructing a formal proof of validity for the following:
1. B<->(L•D), ~D, so ~B
2. R v (S•~T), (R v S) -> (U v ~T), so T->U
3. A -> ([E•~G) ->M], ~ (~E v G) -> (M -> F), so A -> [~ (~G -> ~E) ->F
To construct a formal proof for each statement, we can use logical rules and inference methods. Here's how you can approach each proof:
1. B<->(L•D), ~D, so ~B
Proof:
1. B<->(L•D) (Given)
2. ~D (Given)
3. B -> (L•D) (Biconditional elimination)
4. ~(L•D) (Modus Tollens: 1, 2)
5. ~L v ~D (De Morgan's Law: 4)
6. ~(~L • ~D) (De Morgan's Law: 5)
7. ~~L -> ~D (Implication equivalence rule: 6)
8. L -> ~D (Double negation elimination: 7)
9. ~B (Modus Tollens: 3, 8)
Therefore, ~B is proven.
2. R v (S•~T), (R v S) -> (U v ~T), so T->U
Proof:
1. R v (S•~T) (Given)
2. (R v S) -> (U v ~T) (Given)
3. ~T -> ~(R v S) (Contrapositive: 2)
4. ~T -> (~R & ~S) (De Morgan's Law: 3)
5. T -> (R v S) (Contrapositive: 4)
6. ~(S•~T) -> ~R (Contrapositive: 5)
7. ~R (Assumption)
8. S -> ~T (Modus Tollens: 6, 7)
9. R v S (Disjunction: 1)
10. ~T (Modus Ponens: 8, 9)
11. T -> U (Conditional Proof: 10)
Therefore, T -> U is proven.
3. A -> ([E•~G) ->M], ~ (~E v G) -> (M -> F), so A -> [~ (~G -> ~E) -> F
Proof:
1. A -> ([E•~G) -> M] (Given)
2. ~ (~E v G) -> (M -> F) (Given)
3. ~~ (~G -> ~E) -> F (Double negation elimination)
4. ~ (~G -> ~E) (Assumption)
5. ~F (Modus Ponens: 3, 4)
6. ~((~G -> ~E) -> F) (Negation: 5)
7. ~ (~E v G) (Assumption)
8. ~~E & ~~G (De Morgan's Law: 7)
9. E & ~G (Double negation elimination: 8)
10. [E•~G) -> M (Modus Ponens: 1, Assumption)
11. M (Modus Ponens: 10, 9)
12. ~E v G (Disjunction introduction: 9)
13. M -> F (Modus Ponens: 2, Assumption)
14. F (Modus Ponens: 11, 13)
15. ~~ (~G -> ~E) (Negation: 6)
16. (~G -> ~E) (Double negation elimination: 15)
17. ~G -> ~E (Implication equivalence rule: 16)
18. ~ (~G -> ~E) (Negation: 17)
19. A -> [~ (~G -> ~E) -> F] (Conditional Proof: 18)
Therefore, A -> [~ (~G -> ~E) -> F] is proven.
Remember to study and understand each logical rule and inference method used in the proofs so that you can effectively construct formal proofs in the future.