In standard introductory classes in algebra, trigonometry, and calculus there is currently very lit-tle emphasis on the discipline of proof. set the-ory and recursion theory. The big questions cannot be dodged, and students will not brook a flippant or easy answer. Indirect Proof. proof. 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. The following book is nearly 600 pages long and proceeds at a very slow pace. WUCT121 Logic Tutorial Exercises Solutions 2 Section 1: Logic Question1 … 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. Set theory is also the most “philosophical” of all disciplines in mathematics. 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. Two sets are equal when they have the same elements. Dirk van Dalen, Logic and Structure (Springer, 1994). Some book in proof theory, such as [Gir], may be useful afterwards to complete the information on those points which are lacking. 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 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 early catastrophies, such as the discovery of Relations and Functions . Jon Barwise and John Etchemendy, Language Proof and Logic, 2nd edition (University of Chicago Press, 2003) Common Sets. 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. Predicate Logic 3. . At £41, it is not cheap. Set Theory 5. WUCT121 Logic Tutorial Exercises Solutions 1 WUCT121 Discrete Mathematics Logic Tutorial Exercises Solutions 1. Sn & ÒThe sum of the first n odd numbers is n2.Ó Equivalently, Sn is the statement that: Ò1 + 3 + 5 + (2k-1) + . Logic 2. The notes would never have reached the standard of a book without the 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 Is the 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. Proof is, how-ever, the central tool of mathematics. Questions are bound to come up in any set theory course that cannot be answered “mathematically”, for example with a formal proof. Some very basic knowledge of logic is needed, but we will never go into tedious details. The objects in a set will be called elements of the set. The phenomenon is not unfamiliar, one observes this dichotomy also in other areas, e.g. 2 Sets A set is a collection of objects, which are called elements or members of the set. 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. +(2n-1) = n 2 Ó Assume ÒInductive HypothesisÓ: Sk (foranyparticular k ' 1) 1+3+5+É+ (2k-1) = k2 Add (2k+1) to both sides. The order of the elements in a set doesn't contribute Here are some important sets: Proofs 4. The structure of this proof makes a very convincing demonstration of the validity of the rule of Hypothetical Syllogism. purposes, a set is a collection of objects or symbols. 4 CS 441 Discrete mathematics for CS M. Hauskrecht Equality Definition: Two sets are equal if and only if they have the same elements. This text is for a course that is a students formal introduction to tools and methods of proof. This serves to establish that p was not true to begin with.


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