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. (Caution: sometimes ⊂ is used the way we are using ⊆.) Both signs can be negated using the slash / through the sign.

Because the fundamentals of Set Theory are known to all mathemati-cians, basic problems in the subject seem elementary. Here are three simple statements about sets and functions. They look like they could appear on a homework assignment in an undergraduate course. 1. For any two sets X and Y, either there is a one-to-one function from

I offer no definition of what a set is beyond the intuitive notion described above. Instead, I am going to show you what can be done with sets. This is a typical approach to Set Theory, i.e., sets are treated as primitive s of the theory and are not definable in more basic terms. I adopt the notation in (4) for convenience. (4) a.

Basic set operations De nition Theintersectionof sets A and B is the set of elements in both A and B, denoted A \B := fx jx 2A and x 2Bg: Two sets aredisjointif they have no elements in common, i.e., if A \B = ;. Theunionof sets A and B is the set of elements in either A or B, denoted A [B := fx jx 2A or x 2Bg: Examples

Basic Set Theory John L. Bell I. Sets and Classes. We distinguish between objects and classes. Any collection of objects is deemed to form a class which is uniquely determined by its elements. We write a ∈ A to indicate that the object a is an element or member of the class A. We assume that every member of a class is an object.

such that any other set A is a subset of the universal set, i.e., A ˆU. Example 7. Suppose that U = f1;2;3;:::gis the set of all natural numbers, i.e., all positive integers. The set U would be considered the universal set for Examples 4{6, such that A ˆU and B ˆU for each example. Introduction to Set Theory 3 Nathaniel E. Helwig

that for every set x, there is a set y whose members are exactly the elements of x and x itself, i.e., y..= x ∪{x}. We denote this set y by S(x) and call it the successor of x. Call a set X inductive if ∅∈X and for each x ∈X, its successor S(x) is also an element of X. Infinity axiom states that there exists an inductive set. Axiom 7. Foundation:

The union, the intersection and the di erence of two sets A and B are the sets defined as follows: Union: A [B := fx jx 2A or x 2B g. Intersection: A \B := fx jx 2A and x 2B g. Di We will use the symbol :erence: A B := fx jx 2A and x < Bg.= to define a set. See …

2 Often we use slight variants of the notation above: f3;5;7gis the set con- sisting of 3;5, and 7; more generally, the set consisting of some list of elements is denoted by that list, enclosed in curly brackets, as in fnjn2Nand nis eveng= f0;2;4;:::g; 0:3:1 where again the vertical line jmeans \such that." The symbols are sometimes used backwards; for example, A¾B means