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