Set theory basics Set membership ( ), subset ( ), and equality ( ). ... A proof starts with a list of hypotheses and ends with a conclusion. Set theory is also the most “philosophical” of all disciplines in mathematics. We then present and briefly dis-cuss the fundamental Zermelo-Fraenkel axioms of set theory. We write q 2 X if q is an element. 1.1 Contradictory statements. Then x 2A and y 2B. Sets A set is a collection of things called elements. The objects in a set will be called elements of the set. 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 8 Functions, Bijections, Compositions, Etc.38 The symmetric di erence of A and B is A B = (AnB)[(B nA). A set is a collection of objects, which are called elements or members of the set. These will be the only primitive concepts in our system. Complex issues arise in Set Theory more than any other area of pure mathematics; in particular, Mathematical Logic is used in … The big questions cannot be dodged, and students will not brook a flippant or easy answer. R ˘P(N) Proof. Let sets A, B, and C be given with B C. Then A B = f(a;b) : a 2A^b 2Bg Let (x;y) 2A B. Proof theory was created early in the 20th century by David Hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics| in arithmetic (number theory), analysis and set theory. ELEMENTARY SET THEORY 3 Proof. Questions are bound to come up in any set theory course that cannot be answered “mathematically”, for example with a formal proof. Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 4 Set Theory Basics.doc 1.4. Subsets A set A is a subset of a set B iff every element of A is also an element of B.Such a relation between sets is denoted by A ⊆ B.If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. The negation :(q 2 X) is written as q 2= X. In standard introductory classes in algebra, trigonometry, and calculus there is currently very lit-tle emphasis on the discipline of proof. We can specify a set by listing the elements within braces, Animal = fcat;dog;aardvark;cow;snake;mouse;alligatorg = fdog;dog;aardvark;cat;horse;cow;snake;mouse;alligatorg Note that order and repetitions are irrelevant. We will assume that 2 take priority over everything else. Set operations Set operations and their relation to Boolean algebra. Set Operations and the Laws of Set Theory. This text is for a course that is a students formal introduction to tools and methods of proof. 1. (Caution: sometimes ⊂ is used the way we are using ⊆.) x 2 (X \(Y [Z)) $ x 2 X ^x 2 (Y [Z) x 2 X ^x 2 (Y [Z) $ x 2 X ^(x 2 Y _x 2 Z) English proofs and proof strategies A quick wrap-up of . Set Theory \A set is a Many that allows itself to be thought of as a One." 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. Proof is, how-ever, the central tool of mathematics. 23 (mod5). Since B C, we know y 2C, so it must be that (x;y) 2A C. Thus A B A C. MAT231 (Transition to Higher Math) Proofs Involving Sets Fall 2014 4 / 11 Alternate notation: A B. The union of sets A and B is the set A[B = fx : x 2A_x 2Bg. More sets Power set, Cartesian product, and Russell’s paradox. When expressed in a mathematical context, the word “statement” is viewed in a A set is a collection of objects, called elements of the set. Although Elementary Set Theory is well-known and straightforward, the modern subject, Axiomatic Set Theory, is both conceptually more difficult and more interesting. Two sets are equal when they have the same elements. The proof shows the step-by-step chain of reasoning … Set Theory 2.1.1. (Georg Cantor) In the previous chapters, we have often encountered "sets", for example, prime numbers form a set, domains in predicate logic form sets as well. Sets. A set can be represented by listing its elements between braces: A = {1,2,3,4,5}.The symbol ∈ is used to express that an element is (or belongs to) a set… CHAPTER 2 Sets, Functions, Relations 2.1. We give a proof of one of the distributive laws, and leave the rest for home-work. itive concepts of set theory the words “class”, “set” and “belong to”. By the lemma, it is eanough to show that (0;1) ˘P(N). Complex issues arise in Set Theory more than any other area of pure mathematics; in particular, Mathematical Logic is used in … We make use of the fact that each r2(0;1) has a unique decimal expansion The set di erence of A and B is the set AnB = fx : x 2A^x 62Bg. Nowsuppose n2Z andconsidertheorderedpair(4 ¯3,9 ¡2).Does this ordered pair belong … The intersection of sets A and B is the set A\B = fx : x 2A^x 2Bg. Is the Already in his famous \Mathematical problems" of 1900 [Hilbert, 1900] he raised, as the second We will generally use capital letters for sets. Although Elementary Set Theory is well-known and straightforward, the modern subject, Axiomatic Set Theory, is both conceptually more difficult and more interesting. Likewise,(100,75)2B, (102,77)2B,etc.,but(6,10)ÝB. Theorem 1.20. proof. purposes, a set is a collection of objects or symbols.

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