We will discuss complete set theory ;

A set is a Many that allows itself to be thought of as a One. - Georg Cantor This chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. The material is mostly elementary. For those of you new to abstract mathematics elementary does not mean simple (though much of the material is fairly simple). Rather, elementary means that the material requires very little previous education to understand it. Elementary material can be quite challenging and some of the material in this chapter, if not exactly rocket science, may require that you adjust you point of view to understand it. The single most powerful technique in mathematics is to adjust your point of view until the problem you are trying to solve becomes simple. Another point at which this material may diverge from your previous experience is that it will require proof. In standard introductory classes in algebra, trigonometry, and calculus there is currently very little emphasis on the discipline of proof. Proof is, however, the central tool of mathematics. This text is for a course that is a students formal introduction to tools and methods of proof. 2.1 Set Theory A set is a collection of distinct objects. This means that {1, 2, 3} is a set but {1, 1, 3} is not because 1 appears twice in the second collection. The second collection is called a multiset. Sets are often specified with curly brace notation. The set of even integers can be written: {2n : n is an integer} The opening and closing curly braces denote a set, 2n specifies the members of the set, the colon says “such that” or “where” and everything following the colon are conditions that explain or refine the membership. All correct mathematics can be spoken in English. The set definition above is spoken “The set of twice n where n is an integer”. The only problem with this definition is that we do not yet have a formal definition of the integers. The integers are the set of whole numbers, both positive and negative: {0, ±1, ±2, ±3, . . .}. We now introduce the operations used to manipulate sets, using the opportunity to practice curly brace notation. Definition 2.1 The empty set is a set containing no objects. It is written as a pair of curly braces with nothing inside {} or by using the symbol ∅. As we shall see, the empty set is a handy object. It is also quite strange. The set of all humans that weigh at least eight tons, for example, is the empty set. Sets whose definition contains a contradiction or impossibility are often empty. Definition 2.2 The set membership symbol ∈ is used to say that an object is a member of a set. It has a partner symbol ∈/ which is used to say an object is not in a set. Definition 2.3 We say two sets are equal if they have exactly the same members.and much more.