What are all the maximal subsets of S1 that don’t entail p?
S = Any propositional sentence of A, a maximal subset of S that doesn’t entail A is any set X of propositional sentences.
1. X ⊆ S
2. X ∕|= A
3. For any Y ⊆ S: if X ⊆ Y and Y ⊈ X then Y |= A
Let p, q, r, s be propositional variables and let
S1 = {¬r, s,((¬r ∨ q) → p), q}.
by 
Thecrazyscientist
ANSWERS: 1
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I had ago not sure if this is correct. 5 maximal subsets that do not entail p 1. ¬r ∧ q -> p 2. S2 3. ¬r ∧ S 4. q 5. S1 Subsets 1 -> 4 do not entail p as their members are not mutually exclusive in a way that would lead to the entailing of p. The 5th maximal subset includes q and claims that p is false. All the sets exclude r and s which means that they don't entail either ¬r or s. Taking a set, S, of sentences and X ⊆ S, X = U X which doesn't entail p. S1 = {¬r,s, ((¬r V q) -> p), q} If S1 satisfies 2 conditions, then X ⊆ S1, X ∧ = A and if Y ∩ X Then Y, x can't be false. Therefore if a sentence doesn't entail A then ¬A ⊆ Y Hope this helps 😊
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