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The text contrasts how “truth” and what counts as a proof vary across disciplines, from legal judgments based on presented evidence, to business authority, to experimentally confirmed scientific truth in physics and biology, and to probabilistic conclusions in statistics. It also outlines philosophical proof as careful exposition through small, plausible arguments.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/mathematics-for-computer-science-introduction/571955/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/mathematics-for-computer-science-introduction/571955.png","ImageObject",442,249,{"name":88,"@type":89},"Indoniesian Boy","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":73},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What is the central idea of the text?","Question",{"text":108,"@type":109},"The text centers on using mathematical models and methods to analyze computer science problems, with proof as the key way to establish truth.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How does the document describe “truth” and “proof” across different fields?",{"text":113,"@type":109},"It explains that what counts as proof depends on the discipline—for example, legal systems rely on jury decisions from allowable evidence, while physics and biology rely on experimental confirmation.",{"name":115,"@type":106,"acceptedAnswer":116},"What does the text say about philosophical proof?",{"text":117,"@type":109},"Philosophical proof is presented as careful exposition and persuasion, typically built from a series of small, plausible arguments, with the classic example “Cogito ergo sum.”","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},571955,1792438350,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":73,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":141},962090760608,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","“mcs-ftl” — 2010/9/8 — 0:40 — page i — \\#1  \nMathematics for Computer Science  \nrevised Wednesday 8th September, 2010, 00:40  \nEric Lehman  \nGoogle Inc.  \nF Thomson Leighton Department of Mathematics and CSAIL, MIT  \nAkamai Technologies  \nAlbert R Meyer Massachusets Institute of Technology  \nCopyright © 2010, Eric Lehman, F Tom Leighton, Albert R Meyer . All rights reserved.  \n“mcs-ftl” — 2010/9/8 — 0:40 — page ii — \\#2  \n“mcs-ftl” — 2010/9/8 — 0:40 — page iii — \\#3  \nContents  \nI Proofs  \n1 Propositions 5  \n1.1 Compound Propositions 6  \n1.2 Propositional Logic in Computer Programs 10  \n1.3 Predicates and Quantifiers 11  \n1.4 Validity 19  \n1.5 Satisfiability 21  \n2 Patterns of Proof 23  \n2.1 The Axiomatic Method 23  \n2.2 Proof by Cases 26  \n2.3 Proving an Implication 27  \n2.4 Proving an “If and Only If” 30  \n2.5 Proof by Contradiction 32  \n2.6 Proofs about Sets 33  \n2.7 Good Proofs in Practice 40  \n3 Induction 43  \n3.1 The Well Ordering Principle 43  \n3.2 Ordinary Induction 46  \n3.3 Invariants 56  \n3.4 Strong Induction 64  \n3.5 Structural Induction 69  \n4 Number Theory 81  \n4.1 Divisibility 81  \n4.2 The Greatest Common Divisor 87  \n4.3 The Fundamental Theorem of Arithmetic 94  \n4.4 Alan Turing 96  \n4.5 Modular Arithmetic 100  \n4.6 Arithmetic with a Prime Modulus 103  \n4.7 Arithmetic with an Arbitrary Modulus 108  \n4.8 The RSA Algorithm 113  \n“mcs-ftl” — 2010/9/8 — 0:40 — page iv — \\#4  \niv Contents  \nII Structures  \n5 Graph Theory 121  \n5.1 Definitions 121  \n5.2 Matching Problems 128  \n5.3 Coloring 143  \n5.4 Getting from A to B in a Graph 147  \n5.5 Connectivity 151  \n5.6 Around and Around We Go 156  \n5.7 Trees 162  \n5.8 Planar Graphs 170  \n6 Directed Graphs 189  \n6.1 Definitions 189  \n6.2 Tournament Graphs 192  \n6.3 Communication Networks 196  \n7 Relations and Partial Orders 213  \n7.1 Binary Relations 213  \n7.2 Relations and Cardinality 217  \n7.3 Relations on One Set 220  \n7.4 Equivalence Relations 222  \n7.5 Partial Orders 225  \n7.6 Posets and DAGs 226  \n7.7 Topological Sort 229  \n7.8 Parallel Task Scheduling 232  \n7.9 Dilworth's Lemma 235  \n8 State Machines 237  \nIII Counting  \n9 Sums and Asymptotics 243  \n9.1 The Value of an Annuity 244  \n9.2 Power Sums 250  \n9.3 Approximating Sums 252  \n9.4 Hanging Out Over the Edge 257  \n9.5 Double Trouble 269  \n9.6 Products 272  \n“mcs-ftl” — 2010/9/8 — 0:40 — page v — \\#5  \nv Contents  \n9.7 Asymptotic Notation 275  \n10 Recurrences 283  \n10. 1 The Towers of Hanoi 284  \n10.2 Merge Sort 291  \n10.3 Linear Recurrences 294  \n10.4 Divide-and-Conquer Recurrences 302  \n10.5 A Feel for Recurrences 309  \n11 Cardinality Rules 313  \n11.1 Counting One Thing by Counting Another 313  \n11.2 Counting Sequences 314  \n11.3 The Generalized Product Rule 317  \n11.4 The Division Rule 321  \n11.5 Counting Subsets 324  \n11.6 Sequences with Repetitions 326  \n11.7 Counting Practice: Poker Hands 329  \n11.8 Inclusion-Exclusion 334  \n11.9 Combinatorial Proofs 339  \n11.10 The Pigeonhole Principle 342  \n11.11 A Magic Trick 346  \n12 Generating Functions 355  \n12.1 Definitions and Examples 355  \n12.2 Operations on Generating Functions 356  \n12.3 Evaluating Sums 361  \n12.4 Extracting Coefficients 363  \n12.5 Solving Linear Recurrences 370  \n12.6 Counting with Generating Functions 374  \n13 Infinite Sets 379  \n13.1 Injections, Surjections, and Bijections 379  \n13.2 Countable Sets 381  \n13.3 Power Sets Are Strictly Bigger 384  \n13.4 Infinities in Computer Science 386  \nIV Probability  \n14 Events and Probability Spaces 391  \n14.1 Let's Make a Deal 391  \n14.2 The Four Step Method 392  \n“mcs-ftl” — 2010/9/8 — 0:40 — page vi — \\#6  \nvi Contents  \n14.3 Strange Dice 402  \n14.4 Set Theory and Probability 411  \n14.5 Infinite Probability Spaces 413  \n15 Conditional Probability 417  \n15. 1 Definition 417  \n15.2 Using the Four-Step Method to Determine Conditional Probability 418 15.3 A Posteriori Probabilities 424  \n15.4 Conditional Identities 427  \n16 Independence 431  \n16. 1 Definitions 431  \n16.2 Independence Is an Assumption 432  \n16.3 Mutual Ind","cbCaik1mW6WY790x","https://ap.wps.com/l/cbCaik1mW6WY790x","pdf",3355995,557,"English","# Introduction\n## What counts as a proof across fields\n## Proof as establishing truth in computer science","[{\"question\":\"What is the central idea of the text?\",\"answer\":\"The text centers on using mathematical models and methods to analyze computer science problems, with proof as the key way to establish truth.\"},{\"question\":\"How does the document describe “truth” and “proof” across different fields?\",\"answer\":\"It explains that what counts as proof depends on the discipline—for example, legal systems rely on jury decisions from allowable evidence, while physics and biology rely on experimental confirmation.\"},{\"question\":\"What does the text say about philosophical proof?\",\"answer\":\"Philosophical proof is presented as careful exposition and persuasion, typically built from a series of small, plausible arguments, with the classic example “Cogito ergo sum.”\"}]","Mathematics for Computer Science - Introduction | PDF",1791924701,195]