proof. The intersection of sets A and B is the set A\B = fx : x 2A^x 2Bg. The proof shows the step-by-step chain of reasoning … CHAPTER 2 Sets, Functions, Relations 2.1. Then x 2A and y 2B. 23 (mod5). The objects in a set will be called elements of the set. Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 4 Set Theory Basics.doc 1.4. We write q 2 X if q is an element. 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 1.1 Contradictory statements. When expressed in a mathematical context, the word “statement” is viewed in a Set Theory 2.1.1. 1. ELEMENTARY SET THEORY 3 Proof. These will be the only primitive concepts in our system. 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. Two sets are equal when they have the same elements. In standard introductory classes in algebra, trigonometry, and calculus there is currently very lit-tle emphasis on the discipline of proof. Questions are bound to come up in any set theory course that cannot be answered “mathematically”, for example with a formal proof. Is the A set is a collection of objects, which are called elements or members of the set. By the lemma, it is eanough to show that (0;1) ˘P(N). ... A proof starts with a list of hypotheses and ends with a conclusion. 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) Alternate notation: A B. The negation :(q 2 X) is written as q 2= X. 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 (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. A set is a collection of objects, called elements of the set. Set Operations and the Laws of Set Theory. 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. English proofs and proof strategies A quick wrap-up of . 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 will assume that 2 take priority over everything else. The symmetric di erence of A and B is A B = (AnB)[(B nA). The big questions cannot be dodged, and students will not brook a flippant or easy answer. 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… (Caution: sometimes ⊂ is used the way we are using ⊆.) Complex issues arise in Set Theory more than any other area of pure mathematics; in particular, Mathematical Logic is used in … Sets. Theorem 1.20. More sets Power set, Cartesian product, and Russell’s paradox. itive concepts of set theory the words “class”, “set” and “belong to”. Likewise,(100,75)2B, (102,77)2B,etc.,but(6,10)ÝB. This text is for a course that is a students formal introduction to tools and methods of proof.