Analysis (2 volume set) (Texts and Readings in Mathematics) by Terence Tao

By Terence Tao

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That a+b is always equal to b+a) without even aware that you are doing so; it is difficult to let go and try to inspect this number system as if it is the first time you have seen it. So in what follows I will have to ask you to perform a rather difficult task: try to set aside, for the moment, everything you know about the natural numbers; forget that you know how to count, to add, to multiply, to manipulate the rules of algebra, etc. We will try to introduce these concepts one at a time and identify explicitly what our assumptions are as we go along- and not allow ourselves to use more "advanced" tricks such as the rules of algebra until we have actually proven them.

Because of this, we denote this set as {1, 2} U {2, 3} = {1, 2, 3}. 12. If A, B, A' are sets, and A is equal to A', then A U B is equal to A' U B (why? 4). Similarly if B' is a set which is equal to B, then A U B is equal to A U B'. Thus the operation of union obeys the axiom of substitution, and is thus well-defined on sets. We now give some basic properties of unions. 13. If a and b are objects, then {a, b} ={a} U {b}. , (AUB)UC = AU (BUG)). Also, we have A U A = A U 0 = 0 U A = A. 1. 3. 4, we need to show that every element x of (AUB)U C is an element of A U (B U C), and vice versa.

Also, we have no axioms yet on what sets do, or what their elements do. Obtaining these axioms and defining these operations will be the purpose of the remainder of this section. We first clarify one point: we consider sets themselves to be a type of object. 1 (Sets are objects). If A is a set, then A is also an object. In particular, given two sets A and B, it is meaningful to ask whether A is also an element of B. 2. (Informal) The set {3, {3, 4}, 4} is a set of three distinct elements, one of which happens to itself be a set of two elements.

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