21127 Lecture Notes - Lecture 23: Countable Set, Uncountable Set, Enu
16 Dec 2015
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Extra o ce hours this week: wednesday 2:30-3:30 and 4:30-5:30, thursday 12:00-2:00 and 4:00-6:00. In class on monday, we proved that f is a bijection f is invertible and, furthermore, that when f 1 exists, it is also a bijection and(cid:0)f 1(cid:1) 1. Let"s practice with one example of nding the inversion of a function whose domain/codomain are sets of ordered pairs. Example: consider the function f : r r r r given by. (x, y) r r. f (x, y) = (2x + y, 2y x) Let"s try to nd the inverse of f . The idea is to take an arbitrary (u, v) r r (the codomain) and nd the corresponding (x, y) r r (the domain) that would output that (u, v). This amounts to setting f (x, y) = (u, v) and solving for x and y. This yields a system of two equations in two unknowns.
