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Summary: G53KRR 2009: Solutions to the exercises on first order logic
1. Prove that A1, . . . , An |= B if and only if |= A1 . . . An B
Answer: A1, . . . , An |= B iff
for every interpretation (D, I), if A1, . . . , An are all true in (D, I), then
also B is true in (D, I) iff
for every interpretation (D, I), if A1 . . . An is true in (D, I), then also
B is true in (D, I) iff
for every interpretation (D, I), A1 . . . An B is true in (D, I) (from
the truth definition of ) iff
|= A1 . . . An B.
2. Prove that A1, . . . , An |= B if and only if A1 . . .An ¬B is unsatisfiable.
Answer: A1, . . . , An |= B iff
for every interpretation (D, I), if A1, . . . , An are all true in (D, I), then
also B is true in (D, I) iff
there is no interpretation (D, I) such that A1, . . . , An are all true in (D, I)
and B is false in (D, I) iff
there is no interpretation (D, I) such that A1 . . . An is true in (D, I)
and ¬B is true in (D, I) iff
there is no interpretation (D, I) such that A1 . . . An ¬B is true in
(D, I) iff
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