Correct Answer.
Consider the truth table for the formulas, p, q, 
p 
q, q
| p |
q |
|
|
p q |
q |
| T |
T |
|
|
F |
F |
| T |
F |
|
|
T |
T |
| F |
T |
|
|
T |
F |
| F |
F |
|
|
F |
T |
The lines in which the premises are both true are: line2. In this case the conclusion is also true, so the entailment holds.
Alternatively, to establish p, p q 
q we may reason that this relationship cannot be
violated. Otherwise, there is a valuation which makes q false and the formula p and p
q true. But then, p and q have to be true, so p q then evaluates to
F, contradicting the described situation. Therefore, we cannot find a
valuation that violates the condition for semantic entailment.
(This method may seem more involved than the computation of a truth table,
but it performs much better in practice when formulas get bigger.)
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