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Denumerable Sets

We know that is the smallest infinite set; this makes a pretty good place to get started. But our ideas about size have to do...

Finite Sets and Infinite Sets

Grab some collection of objects and count it. How does that work? Let's say we want to count the (living) former Presidents of the United...

Functions whose Inverses are Functions

Recall that we defined inverses for any relation: Definition. If is a relation from to , then is a relation from to , defined by...

Sets of Functions

Consider the set of all functions . Each subset has such a function, namely its characteristic function, ; moreover, given we can define the set...

Functions and Sets

Pushforwards and Pullbacks A function gives us a way of taking elements in and pushing them over to . We call the image of under...

Operations on Functions

Set Operations on Functions Functions are sets, so we can try to combine them like we combine sets. That is, we can ask the following...