Here are my answers to Stanford Introduction To Logic: 3.5.

In each of the following cases, determine whether the given individual sentence is consistent with the given set of sentences.

a. {p ∨ q, p ∨ ¬q, ¬p ∨ q} and (¬p ∨ ¬q)

Using a truth table:

p q p ∨ q p ∨ ¬q ¬p ∨ q ¬p ∨ ¬q

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T T T T T F F T T F T T T F T T F T F F F T T T

So no, the sentence isn't consistent.

b. {p ⇒ r, q ⇒ r, p ∨ q} and r

p q r p ⇒ r q ⇒ r p ∨ q r

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T T T T T T T F T T T T T T T F T T T T T F F T T T F T T T F F F T F F T F T F T F T F F F T T F F F F T T F F

The first interpretation satisfies the formula, so yes it is consistent

c. {p ⇒ r, q ⇒ r, p ∨ q} and ¬r

p q r p ⇒ r q ⇒ r p ∨ q ¬r

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T T T T T T F F T T T T T F T F T T T T F F F T T T F F T T F F F T T F T F T F T T T F F F T T T F F F T T F T

The formula is unsatisfiable, so not consistent.

d. {p ⇒ q ∨ r, q ⇒ r} and p ∧ q

p q r p ⇒ q ∨ r q ⇒ r p ∧ q

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T T T T T T F T T T T F T F T T T F F F T T T F T T F T F T F T F T T F T F F F F F F F F T T F

The formula is satisfiable, so consistent.

e. {p ⇒ q ∨ r, q ⇒ r} and q ∧ r

p q r p ⇒ q ∨ r q ⇒ r p ∧ q

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T T T T T T F T T T T F T F T T T F F F T T T F T T F T F T F T F T T F T F F F F F F F F T T F

The formula is satisfiable, so consistent.