Show that the argument form with premises $(p∧t)→...¬s$ and conclusion $q→r$ is valid

Question

Stuck on this problem. I want to use the rules of inference to show that the argument form with premises $(p∧t)→(r∨s)$, $q→(u∧t),u→p$, and $¬s$ and conclusion $q→r$ is valid. Would really appreciate if someone can help me solve it and explain which rules they used

Answer 1

$\neg s$ premise $1$

$(p \wedge t) \to (r \lor s)$ premise $2$

$(p \land t) \to r$ based on $1$ and $2$ Disjunctive Syllogism

$q \to (u \land t)$ premise $4$

$q \to r$ (hypothetical syllogism) from $3$ and $4$

Is this answer correct? I am not sure if $(p \land t) == (u \land t)$. Because based on Hypothetical syllogism, $P \to Q$ and $Q \to R \implies P \to R$. In this cause I am taking $(p \land t) == (u \land t) == Q$

Answer 2

$(p∧t)→(r∨s), q→(u∧t), u→p, ¬s \vdash q→r$

$ \begin{array}{11111} \{1\} & 1. & (p∧t)→(r∨s) & \text{premise} \\ \{2\} & 2. & q→(u∧t) & \text{premise} \\ \{3\} & 3. & u→p & \text{premise} \\ \{4\} & 4. & ¬s & \text{premise} \\ \{5\} & 5. & q & \text{Assumption for Conditional Proof} \\ \{2,5\} & 6. & u∧t & \text{2,5 Modus Ponens} \\ \{2,5\} & 7. & u & \text{6 Conjuntion Elimination} \\ \{2,5\} & 8. & t & \text{6 Conjuntion Elimination} \\ \{2,3,5\} & 9. & p & \text{3,7 Modus Ponens} \\ \{2,3,5\} & 10. & p \wedge t & \text{8,9 Conjunction Introduction} \\ \{1,2,3,5\} & 11. & r \vee s & \text{1,10 Modus Ponens} \\ \{1,2,3,4,5\} & 12. & r & \text{4,11 SI: Disjunctive Syllogism} \\ \{1,2,3,4\} & 13. & q \to r & \text{5,12 Conditional Proof} & \square \\ \end{array} $

Answer 3

$ 1) (p\land q)\to (r\lor s) :Premise $

$ 2) \lnot p\lor \lnot t \lor r\lor s : $ Logical equivalence of (1)

$ 3) s \lor (\lnot p \lor \lnot t \lor r) : $ Commutative and associative laws on (2)

$ 4) \lnot s : Premise $

$ 5) \lnot p \lor (\lnot t \lor r) : $ Disjunctive syllogism on (3) and (4)

$ 6) u \to p : Premise $

$ 7) \lnot u \lor p $ Logical equivalence of (6)

$ 8) p \lor \lnot u : $ Commutative law on (7)

$ 9) \lnot u \lor \lnot t \lor r : $ Resolution law on (5) and (8)

$ 10) \lnot(u \land t) \lor r : $ De Morgan's law on (9)

$ 11) (u \land t) \to r : $ Logical equivalence of (10)

$ 12) q \to (u \land t) : $ Premise

$ 13) q \to r : $

Hypothetical syllogism on (12) and (11)