Introduction to Logic

Exercise 3.4 - Propositional Entailment

Let Γ and Δ be sets of sentences in Propositional Logic, and let φ and ψ be individual sentences in Propositional Logic. State whether each of the following statements is true or false.

a. If Γ ⊨ φ and Δ ⊨ φ, then Γ ∩ Δ ⊨ φ.
b. If Γ ⊨ φ and Δ ⊨ φ, then Γ ∪ Δ ⊨ φ.
c. If Γ ⊨ φ and Δ ⊭ φ, then Γ ∪ Δ ⊨ φ.
d. If Γ ⊭ ψ, then Γ ⊨ ¬ψ.
e. If Γ ⊨ ¬ψ, then Γ ⊭ ψ.