This text is for a course that is a students formal introduction to tools and methods of proof. Proofs 4. An Overview of Logic, Proofs, Set Theory, and Functions aBa Mbirika and Shanise Walker Contents 1 Numerical Sets and Other Preliminary Symbols3 2 Statements and Truth Tables5 3 Implications 9 4 Predicates and Quanti ers13 5 Writing Formal Proofs22 6 Mathematical Induction29 7 Quick Review of Set Theory & Set Theory Proofs33 The big questions cannot be dodged, and students will not brook a flippant or easy answer. A special case of Conditional Proof is to assume p and then reach as a contradiction the conjunction of q and ~ q for some sentence q. Predicate Logic 3. Questions are bound to come up in any set theory course that cannot be answered “mathematically”, for example with a formal proof. Logic appears in a ‘sacred’ and in a ‘profane’ form; the sacred form is domi-nant in proof theory, the profane form in model theory. Some very basic knowledge of logic is needed, but we will never go into tedious details. Is the purposes, a set is a collection of objects or symbols. Indirect Proof. 1+3+5+É+ (2k-1)+(2k+1) = k2 +(2k+1) Sum of first k+1 odd numbers= (k+1)2 CONCLUDE: Sk+1 Sn & ÒThe sum of the first n odd numbers is n 2.Ó The objects in a set will be called elements of the set. . Dirk van Dalen, Logic and Structure (Springer, 1994). The structure of this proof makes a very convincing demonstration of the validity of the rule of Hypothetical Syllogism. Some early catastrophies, such as the discovery of Sets are usually described using "fg" and inside these curly brackets a list of the elements or a description of the elements of the set. The notes would never have reached the standard of a book without the Some book in proof theory, such as [Gir], may be useful afterwards to complete the information on those points which are lacking. set theory is a theory of pure well-founded sets and its intended models are structures of the form hR( );2i, where the numbers will depend upon the particular axioms included in the theory. proof. At £41, it is not cheap. The phenomenon is not unfamiliar, one observes this dichotomy also in other areas, e.g. In standard introductory classes in algebra, trigonometry, and calculus there is currently very lit-tle emphasis on the discipline of proof. The order of the elements in a set doesn't contribute This serves to establish that p was not true to begin with. Example: • {1,2,3} = {3,1,2} = {1,2,1,3,2} Note: Duplicates don't contribute anythi ng new to a set, so remove them. A )(B )C) (A and B) )C conditional proof In a course that discusses mathematical logic, one uses truth tables to prove the above tautologies. set the-ory and recursion theory. Two sets are equal when they have the same elements. Logic 2. Common Sets. 2 Sets A set is a collection of objects, which are called elements or members of the set. Sn & ÒThe sum of the first n odd numbers is n2.Ó Equivalently, Sn is the statement that: Ò1 + 3 + 5 + (2k-1) + . Jon Barwise and John Etchemendy, Language Proof and Logic, 2nd edition (University of Chicago Press, 2003) 4 CS 441 Discrete mathematics for CS M. Hauskrecht Equality Definition: Two sets are equal if and only if they have the same elements. WUCT121 Logic Tutorial Exercises Solutions 2 Section 1: Logic Question1 … The following book is nearly 600 pages long and proceeds at a very slow pace. Here are some important sets: Set Theory 5. WUCT121 Logic Tutorial Exercises Solutions 1 WUCT121 Discrete Mathematics Logic Tutorial Exercises Solutions 1. Set theory is also the most “philosophical” of all disciplines in mathematics. Proof is, how-ever, the central tool of mathematics. Relations and Functions . Logic andSet Theory Lectured by I.B.Leader, LentTerm 2005, 2010 Chapter 1 Propositional Logic 1 Chapter 2 Well-Orderings and Ordinals 7 Chapter 3 Posets and Zorn’s Lemma 16 Chapter 4 Predicate Logic 24 Chapter 5 Set Theory 34 Chapter 6 Cardinals 43 Bonus lecture Incompleteness Examples Sheets Prerequisites.


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