14. Use a direct proof to show that the following argument is valid. Need to add

Question

14.

Use a direct proof to show that the following argument is valid.

Need to add the lines to finish the proof. The first box should be the proof, second two boxes should reference the line numbers, the last box should say which rule we are using. Thank you in advance!!

4 VCheck DeMorgan's Rule DM Commutativity Com Modus Ponens MP Modus Tollens MT Hypothetical Syllogism HS Associativity Assoc Disjunctive Syllogism DS Constructive Dilemma CD Double Negation DA Simplification Simp Conjunction Conj Addition Add Add Line X Delete Line Check Proof Distribution Dist

Explanation / Answer

1. ~(I V J)

2. ~(I V M)? (E ? W)

3. ~(J V M)? (W ? N)

To Prove : E ? N

4. ~ I . ~J 1, DeMorgans Law (1, DM)( . stands for AND operation)

a. By Simplification law we know:

p.q = p

5. ~ I . ~ J = ~I (4, Simpl)

b. By Addition Law:

p = pV q

So apply Demorgans law to '2' and then apply addition law

6. (~ I . ~ M) ? (E ? W) (2, DM)

7. ~ I ? (E ? W) (6, Addition Law)

Now we can use Implication Law that states p ? q= ~p V q

8. I V (E? W) (7, Impl)

We know p.q=q.p

Using Commutative Law:

9. (E? W) V I (8, Comm)

Again Using Addition law,

10. E? W (9, Addn)

From Same explanation above we will deduce W ? N

11. (~ J . ~ M) ? (W ? N) (2, DM)

12. ~ J ? (W ? N) (11, Addition Law)

13. J V (W ? N) (12, Impl)

14. (W? N) V J (13, Comm)

15. W ? N (14, Add)

So till now we got

E? W and W ? N . Based on the Hypothetical Syllogism rule, iff p ? q and q ? r , then p ? r

So,

16. E ? N ( 10, 15, Hypothetical Syllogism)

Hence Proved

I have tried to give entire steps with full explanation how each step is transformed to get the final answer. I have used Numbers 1 to 16 to label each of the statements. Whatever is written in bracket will be coming in boxes 2, 3 and 4 as you have demanded

I hope my explanation would give you full insight of what all steps we are using to prove the statement

Thankyou

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