Proof, sets, and logic department of mathematics boise state. This chapter will be devoted to understanding set theory, relations, functions. Issues about data structures used to represent sets and the computational cost of set operations. In fact, this is the only rigorous way to prove that two sets. Many different systems of axioms have been proposed. Discrete mathematics, set theory, and logic discrete math, graph theory, combinatorics, recurrence relations difference equations, set theory, gene. These notes provide a very brief background in discrete mathematics. Continued since each set is a subset of the other, we have established the equality of the two sets so a b c a b \a c. Set theory basic building block for types of objects in discrete mathematics. A set is an unordered collection of distinct objects. For example, the set 2,4,17,23 is the same as the set 17,4,23,2. Proofs are to mathematics what spelling is to poetry. In earlier versions of our logic style manual, we tended to.
An important feature of a set is that its elements are \distinct or \uniquely identi able. The book began as a set of notes for the discrete mathematics course at the. Math 522, advanced set theory, at boise state university, on the practical level. Basic set theory members of the collection comprising the set are also referred to as elements of the set. Definitions and the element method of proof recall that a set is a collection of elements. Learning how to write mathematical proofs takes time and hard work. Klostermeyer school of computing university of north florida jacksonville, fl 32224 email. A set is typically determined by its distinct elements, or members, by. The sets shown to exist by this theorem are in a sense discrete they. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematical proofs.
This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. An overview of logic, proofs, set theory, and functions. This alone assures the subject of a place prominent in human culture. Mat231 transition to higher math proofs involving sets fall 2014 8 11. Elements of a set can be just about anything from real physical objects to abstract mathematical objects. But even more, set theory is the milieu in which mathematics takes place today. Sets we discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. As with any mathematical proof, the first step is to try to get a handle on how all the pieces move. The objects in a set are called the elements, or members, of the set. We can list each element or member of a set inside curly brackets like this.