Conditional Proof

Question

Please Show the solution the premise and the laws that you used i can't get it thanks number two is the most i need to have an answer .

1.)

     1. (M v N) → O

     2. P→(Q ^ R) /∴ (M → O) → (P → Q)

 

 

2.)

A→B /∴(A ^ C) → (B ^ C)

Answer 1

I will focus on question 2.

Given A → B show that (A ∧ C) → (B ∧ C). I am using & for logical AND,

Now A → B = ¬A ∨ B. Hence ¬A ∨ B is true. This means at least one of ¬A and B is true.

Case 1. ¬A is true.

=> ¬A ∨ ¬B is true.

=> ¬(A ∧ B) is true since ¬A ∨ ¬B = ¬(A ∧ B)

=> ¬(A ∧ C) ∨ (B ∧ C) since one part of OR is true.

QED Since ¬(A ∧ C) ∨ (B ∧ C) = (A ∧ C) → (B ∧ C).

Case 2. B is true.

Case 2.1 C is true.

=> B ∧ C is true since B and C are both true.

=> ¬(A ∧ C) ∨ (B ∧ C) since one part of OR is true.

QED Since ¬(A ∧ C) ∨ (B ∧ C) = (A ∧ C) → (B ∧ C).

Case 2.1 C is false.

=> A ∧ C is false since A and C are both not true.

=> ¬(A ∧ C) is true.

=> ¬(A ∧ C) ∨ (B ∧ C) since one part of OR is true.

QED Since ¬(A ∧ C) ∨ (B ∧ C) = (A ∧ C) → (B ∧ C).

QED Since we proved the statement in each case.