MATH 0050: Logic Exercise Set A 2022-2023
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MATH 0050: Logic
2022-2023
Exercise Set A
1) Consider the following strings in Lstring (for variable symbols x, y, a unary predicate symbol P and a binary predicate symbol Q):
¬∀yPy ⇒⇒ Qxy ¬Px ; ⇒ ¬∀yPy ⇒ Qxy ¬Px
For each of the above strings, determine, with justification, whether or not it is a formula in L.
2) Convert the following formula in Lmath to an equivalent formula in L (for variable symbols x, y, a unary predicate symbol P and a binary predicate symbol Q):
((Px) ∨ (¬ (Qxy))) ⇒ (((∀x)(Px)) ∧ (Qyx))
3) Consider the following proposition (where α, β , γ are taken to be distinct primitive propositions):
(α ⇒ ((¬β) ∧ γ)) ⇒ (α ⇒ (¬γ))
Use the semantic tableaux method to determine whether or not the given proposition is a tautology; if the proposition is not a tautology, describe every (type of) valuation for which it fails to be true.
4) Consider the following (where α, β , γ are taken to be distinct primitive propositions):
{α ⇔ (β ∨ γ)} |= (¬α) ⇒ (γ ⇒ β)
Use the semantic tableaux method to determine whether or not the given is a semantic implication that holds; if it does not hold, describe every (type of) valuation for which it fails to hold.
2023-02-18