SOLUTION: p ↔ (p v q ) prove:q → p

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Question 935859: p ↔ (p v q )
prove:q → p
Found 2 solutions by Theo, Edwin McCravy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
you set up a truth table for p, q, (p v q) and p <-> (p v q)

(p v q) is true if one of them is true.
(p v q) is false if both are false.

p <-> (p v q) is true if both have the same truth value.
p <-> (p v q) is false if both do not have the same truth value.

q -> p is false if q is true and p is false.
otherwise q -> p is true.

if the truth table for p <-> (q v p) is has exactly the same truth values as the truth table for q -> p, then they are equivalent.

the truth tables are shown below:

$$$
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
I'm sure your teacher didn't mean to use truth tables.

1. p ↔ (p v q )
       prove:q → p

2. (p v q) → p      1,BE biconditional simplification, elimination

3. ~p → ~(p v q)    2,MT modus tollens

4.  ~p → (~p & ~q)  3,DM DeMorgan's law 

5.  ~p → ~q         4,CS conjunctive simplification

6.  ~~q → ~~p       5,MT modus tollens

7.  q → p           6,DN double negation 

Edwin
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