Gust the blue solve ....... Discerte math

Construct a truth table for each of these compound propositions. a)p -p b) p -p c) p (pva) d) (p Aq) (PV) Question 5: Express these system specifications using the propositions p "The message is scanned for viruses" and a "The message was sent from an unknown system" together with logical connectives (including negations) a) "The message is scanned for viruses whenever the message was sent from an unknown system." b) "The message was sent from an unknown system but it was not scanned for viruses." c) "It is necessary to scan the message for viruses whenever it was sent from an unknown system." d) "When a message is not sent from an unknown system it is not scanned for viruses." Question 6: Are these system specifications consistent? "Whenever the system software is being upgraded, users cannot access the file system. If users can access the file system, then they can save new files. If users cannot save new files, then the system software is not being upgraded." Question 7 Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output ((p V-r)A-9) V-PA (Vr)) from input bits p, q, and r. Section 1.3: Question 8 Use a truth table to verify the first De Morgan law -( p a) - V-q. Suppose that the domain of the propositional function P(x) consists of the integers 1, 2, 3, 4, and 5. Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions. a) 3xP(x) b) VxP(x) c) -3xP(x) d) -VxP(x) e) Vx[(x = 3) P(x)) V 3x-P(x) Question 17: Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. a) Something is not in the correct place. b) All tools are in the correct place and are in excellent condition. c) Everything is in the correct place and in excellent condition. d) Nothing is in the correct place and is in excellent condition. e) One of your tools is not in the correct place, but it is in excellent condition