Let Γ and Δ be sets of sentences in Relational Logic. Let φ and ψ be closed sentences in Relational Logic. State whether each of the following statements is true or false.
a.
If Δ ⊨ ∀x.(p(x) ∨ q(x)) and Δ ⊨ ∃x.¬p(x), then Δ ⊨ ∃x.q(x)
b.
If Δ ⊨ ∀x.(p(x) ⇒ q(x)) and Δ ⊨ ∃x.p(x), then Δ ⊨ ∃x.(p(x) ∧ q(x)).
c.
If Δ ⊨ ∀x.(p(x) ⇒ q(x)) and Δ ⊨ ∀x.(q(x) ⇒ r(x)), then Δ ⊨ ∀x.(p(x) ⇒ r(x)).
d.
If Δ ⊭ ∃x.p(x), then Δ ⊨ ∀x.¬p(x).
e.
If Δ ⊨ ∀x.(p(x) ⇒ q(x)) and Δ ⊨ ∃x.¬q(x), then Δ ⊨ ∃x.¬p(x).
