Most of the period was occupied by the presentation of the proofs of the basic property of ordered pairs and the existence of cartesian products. At the end, I gave some definitions relevant to our enterprise of ``enumeration''. Definition: A class of ordered pairs is a ``binary relation''. Observation: any infix predicate R on sets can be coded by the class (not necessarily set!) {(x,y) : x R y}. Definition: The ordered triple (x,y,z) is defined as ((x,y),z). Definition: For sets A and B, a relation from A to B is an ordered triple (A,R,B) such that R c= A x B. A is said to be the domain of the relation and B is said to be its codomain (or sometimes, speaking loosely, its range, though the definition of range is a little different). Convention: the relation from A to B formally represented by (A,R,B) will often be referred to as just R, where A and B are understood in context. Convention: the locutions x R y and (x,y) E R will be regarded as synonymous where this does not lead to confusion (an infix predicate will be confused with the class (or set) that codes it). Observation: Notice that the relation R which is a component of a relation from A to B is actually a set. Definition: A relation from A to B, (A,R,B), is said to be ``source covering'' iff (Ax E A.(Ey E B.(x R y))). (A,R,B) is said to be ``source constricted'' iff (Axy E A.(Az E B.((x R z & y R z) -> x = y))). A relation from A to B, (A,R,B), is said to be ``target covering'' iff (Ax E B.(Ey E A.(y R x))). (A,R,B) is said to be ``target constricted'' iff (Axy E B.(Az E A.((z R x & z R y) -> x = y))). This can best be understood using a metaphor. Think of all elements of A as archers and elements of B as targets. x R y iff x fires an arrow which hits y. ``Source covering'' is pictured as ``every archer fires at least one arrow''. ``Source constricted'' is pictured as ``every archer fires at most one arrow''. ``Target covering'' is pictured as ``every target is hit by at least one arrow''. ``Target constricted'' is pictured as ``every target is hit by at most one arrow''. It is logically interesting that the concepts of ``at least one'' and ``at most one'' can be captured entirely in logical notation. Notice that this means that ``exactly one'' can be captured as well! Definition: A relation (A,R,B) from A to B is a function from A to B just in case it is source covering and source constricted. Observation: In terms of our metaphor, this means that each archer fires exactly one arrow. Definition (function notation): For any function (A,f,B) from A to B (usually referred to as just f) and any element x of A, we define f(x) as the uniquely determined y such that (x,y) E f. Definition: A function which is ``target constricted'' is said to be one-to-one or said to be an injection. Definition: A function which is ``target covering'' is said to be onto or a surjection. Observation: Notice that if we think of a function just as a set of ordered pairs, we cannot tell whether it is a surjection or not. Definition: A function which is both a surjection and an injection (one-to-one and onto) is said to be a bijection. Definition: Sets A and B are said to be the same size, or to be equinumerous (A ~=~ B) just in case there is a bijection (A,f,B). Observation: the definition of relation from A to B can actually be extended safely to allow A and B to be classes all of whose members are sets, and using this extension it is possible to define A ~=~ B for any classes of sets. The definition could be extended to even more general classes, but it would become less meaningful the farther it was extended... Definition: A set A is said to be finite just in case there is a natural number n such that A ~=~ n. In this case, we would like to say |A| = n, but there is a theorem to be proved to the effect that there is at most one natural number n such that A ~=~ n. This is where we stopped on April 7. A very important intuitive issue which has been raised by a student has to do with whether sets can be infinite. The short answer is that we have proved that there is an infinite set (Theorem 7), but the question deserves a longer answer. The distinction between sets and classes does have to do with the idea of infinity; classes are infinite in a more unmanageable way than sets. Arbitrary collections of classes, which we do not allow ourselves to talk about, are even worse!