Question1:
A.
1) false
2) false
B.
Leave to you
C.
1) 6 2)4 3)4
Question2:
A.
(a) A valid sentence is one that is true in all models. The sentence True
is also valid in all models. So if αis valid then T rue |=αholds (because
both True and αhold in every model), and if the entailment holds then α
must be valid, because it must be true in all models in which True holds.
(b) False doesn’t hold in any model, so αtrivially holds in every model
that False holds in.
(c) Assume α|=β. Then consider any model m. If αis true in m, then
by our assumption, βis also true in m, so α⇒βis true in m. Otherwise,αis
false in m, so α⇒βis true in m. Thus α⇒βis valid. Conversely, suppose
α⇒βis valid. Then consider any model mwhereαis true. βmust also be
true in m, or else mwould not satisfy α⇒β. So α|=β.
(d) This reduces to c, because α∧ ¬βis unsatisable just when α⇒βis
valid.
B.
1) VALID. By truth table: both F alse ⇒F alse and Tr ue ⇒T rue are
true.
2) SATISFIABLE. By truth table: it is true when Smoke is true and
F ire is true, but false when S moke is false and F ir e is true.
3) VALID. By truth table: if Smoke is false, both expressions are false;
if Smoke is true then either the (Smoke ⇒F ire) is false (in which case the
whole implication is true) or F ire is true, in which case both sides are true.
4) SATISFIABLE. By truth table: true when both Big and Dumb are
true, but false when Big is false and and Dumb is true.
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