{"id":1873,"date":"2024-07-19T08:16:18","date_gmt":"2024-07-19T08:16:18","guid":{"rendered":"https:\/\/britainwriters.com\/answers\/?p=1873"},"modified":"2024-07-19T08:16:21","modified_gmt":"2024-07-19T08:16:21","slug":"tautology-and-contradiction-assignment","status":"publish","type":"post","link":"https:\/\/britainwriters.com\/answers\/tautology-and-contradiction-assignment\/","title":{"rendered":"Tautology And Contradiction Assignment"},"content":{"rendered":"\n<h3 class=\"wp-block-heading\"><strong>Assignment Task<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 1<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A tautology is a statement that is true for every valuation of the propositional variables it contains; a contradiction is a statement that is false for every valuation of the variables; and a contingency is a statement that is neither a tautology nor a contradiction.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Which of the following are tautologies, which are contradictions, and which are contingencies?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(a) p&nbsp;\u27be (pV ~p)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(b) (p\u27be q)&nbsp;\u21d4(q\u27be&nbsp; 7p)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(c) (p\u028cq)&nbsp;\u27be r)\u21d4(r\u27be (p\u028cq))<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In each case, you should present a truth table to justify your answer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 2<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using the algebra of propositional logic, rewrite the propositions of Question 1 in their simplest equivalent forms. (By simplest, we mean the form&nbsp;that has the fewest logical connectives.) In each case you should justify the&nbsp;rewriting through a chain of equivalences of the form<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">p=&gt; (pV -p)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 3<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By constructing proof trees, show that the tautologies of Question 1 (if there<br>are any) are theorems of the propositional calculus. In doing so, you may<br>make use of any theorems proved during the course.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 4<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Assume two sets, A = {0,1,2} and B = {0,1}. Assume also a relation,<br>R\u2208 A&nbsp;\u27f7 B, such that R= {0\u27f71,1\u27f70,2\u27f71}.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(a) Give values for the following.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(i) #(A \u00a9 B)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(ii) #(A + B)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(iii) #(A &gt; B)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(iv) #(A +\u00bb B)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(v) #(A \u2014 B)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(vi) #(A &gt; B)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(b) Give values for the following.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(i) The reflexive closure of R.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(ii) The symmetric closure of R.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(iii) The transitive closure of R.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 5<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The relational composition operator is associative, which is to say that, if<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">R \u2208 W \u27f7 X, S \u2208 X&nbsp;\u27f7 Y and T&nbsp;\u2208 Y&nbsp;\u27f7 Z, then the statement<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(R:S):T = R: (S:T)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is a theorem. To prove this, it is enough to complete the following chain of<br>equivalences:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">w \u27f6 z \u20ac (R:S) :T<br>=&gt;<br>w \u27f6 z \u20ac R :(S:T)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Complete this (short) proof by completing the chain, justifying each step<br>with a reference to a standard definition or an equivalence of predicate logic.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 6<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">You have been asked to think about a data model for an initial prototype of&nbsp;a messaging system.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The model makes use of the following basic types, free types and abbreviations:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[ User Id ]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">| Messageld |<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">YesNo := Yes | No<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Timestamp == N<br>The variable current represents the current time:<br>| current : Timestamp<br>The data associated with the system is represented via the following.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">users : F Userld<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">messages : MessageId ++ (UserId x Userld x Timestamp)<br>received_status : MessageId + YesNo<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">received_time : Messageld ++ Timestamp<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">read_status : MessageId ++ YesNo<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">read_time : Messageld ++ Timestamp<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>This captures the following:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The system keeps track of a (finite) set of users.<\/li>\n\n\n\n<li>The messages function maps message identifiers to triples. In position 1 of each such triple is the identifier of the user who sent the message; in position 2 is the identifier of the recipient; in position 3 is the time at which it was sent.<\/li>\n\n\n\n<li>The functions received_status and read_status are concerned with the status of messages whether they\u2019ve been read or received, respectively.<\/li>\n\n\n\n<li>The functions received_time and read_time document the times at which messages are received and read, respectively.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Complete the definition by writing the following predicates.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(a) Any user identifier that appears in messages must also appear in users.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(b) Users can\u2019t send messages to themselves.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(c) All recorded messages must have been sent before the current time.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(d) Only sent messages can be received.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(e) Only received messages can be read.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(f) All received messages (and no others) appear in received_time.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(g) All read messages (and no others) appear in read_time.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(h) Messages can be received only after they\u2019ve been sent.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(i) Messages can be read only after they\u2019ve been received.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 7<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Assuming the sets and functions of the previous question, represent the following via the mathematical language of Z.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(a) The set of identifiers of messages that have been sent, but not received.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(b) The set of identifiers of messages that have been received, but not read.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(c) The identifier of the user who sent the first message. (You should feel<br>free to assume that there is a unique such user.)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(d) The identifier of the user who has sent the most messages. (You should<br>feel free to assume that there is a unique such user.)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 8<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Consider the following free type definition:)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Status := SentButNotReceived | ReceivedButNotRead | Read<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">a) Assuming the above, together with the sets and functions of Questions 6&nbsp;and 7, define \u2014 in terms of messages, read_status and received_status&nbsp;(and anything else that you deem to be of relevance) \u2014 a function,&nbsp;message_status \u20ac Messageld ++ Status, that maps message identifiers&nbsp;to their status. So, for example, given some m \u20ac Messageld as in-put, the function would return ReceivedButNotRead as output if that<br>message has been received, but not read.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">b) Assuming the above, together with the sets and functions of Questions 6&nbsp;and 7, define \u2014 in terms of messages, read_status and received_status&nbsp;(and anything else that you deem to be of relevance) \u2014 a function<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">change_status_to_read \u20ac<br>((MessageId ++ Status) x MessageId) +\u00bb<br>(Messageld ++ Status)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">that has the effect of changing the status of a specific message from<br>ReceivedButNotRead to Read.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 9<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Assume the existence of message_status:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">| message_status : Messageld ++ Status<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">a) Define a recursive function, count, which takes as input a sequence,s \u20ac seq Messageld, and returns as output the number of elements that appear in s.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">b) Define a recursive function, not_received, which takes as input a sequence, s \u20ac seq Messageld, and returns a subsequence of s containing all the messages of whose status is SentButNotReceived (according to message_status).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">c) Define a recursive function, not_read, which takes as input a sequence, s \u20ac seq Messageld, and returns a subsequence of s containing all the messages of s whose status is ReceivedButNotRead (according tomessage_status).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">d) Define a recursive function, read, which takes as input a sequence,s \u20ac seq Messageld, and returns a subsequence of s containing all the messages of s whose status is Read (according to message_status).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">e) Show, via structural induction, that<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Vs:seq MessageId&nbsp;<br>count (not_received (s))<br>+<br>count (not_read (s))<br>+<br>count (read (s))<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">count (s)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question 10<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Consider the following free type definition.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">MessageList :=<br>nil |<br>join ( Messageld x UserId x UserId x Timestamp x Status x<br>MessageList ))<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(a) Give a suitable induction principle for this free type.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(b) Define recursive functions count,, not_received,, not_read;,, and read;&nbsp;that take as input elements of MessageList and have behaviour that&nbsp;correspond to the functions of Question 9.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(c) Show, via structural induction, that<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Vi: MessageList e<br>county, (not_receivedy, (1))<br>+<br>count, (not_read,, (1))<br>+<br>count, (read, (1))<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">count, (1)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Assignment Task Question 1 A tautology is a statement that is true for every valuation of the propositional variables it contains; a contradiction is a statement that is false for every valuation of the variables; and a contingency is a statement that is neither a tautology nor a contradiction. Which of the following are tautologies, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1873","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Tautology And Contradiction Assignment - My Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/britainwriters.com\/answers\/tautology-and-contradiction-assignment\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Tautology And Contradiction Assignment - My Blog\" \/>\n<meta property=\"og:description\" content=\"Assignment Task Question 1 A tautology is a statement that is true for every valuation of the propositional variables it contains; a contradiction is a statement that is false for every valuation of the variables; and a contingency is a statement that is neither a tautology nor a contradiction. 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