a) Write the following as English sentences where F(p) is "Printer p is out of service,"B(p) is "Printer p is busy," Lj) is "Printer job j is lost", and Q() is "Printer job j is queued." \text { i. } \quad \exists p(F(p) \wedge B(p)) \rightarrow \exists j L(j) \text { ii. } \quad \forall p B(p) \rightarrow \exists j Q(j) \text { iii. } \quad \exists j(Q(j) \wedge L(j)) \rightarrow \exists p F(p) b) Use Logical Equivalence Theorem to verify the logical equivalences below. (p \wedge(\sim(\sim p \vee q))) \vee(p \wedge q) \equiv p c) A set of premises and a conclusion are given below. Use the valid arguments forms to deduce the conclusion from the premises, giving a reason for each step. Assume all variables are statement variables. \sim p \vee q \rightarrow r

p \rightarrow t \sim p \wedge r \rightarrow \sim s d) Determine the truth value of each of the following statements if the domain for all variables consists of all integers. \text { i. } \forall n \exists m\left(n^{2}