Jerusalem College of Technology: Dr. Dana-Picard's "The Characteristic Function of a Set"
Read this page. This brief text will show you how to use characteristic functions to prove properties of sets. However, there are other reasons to learn how to do this. You will see later in the course that functions (not just characteristic functions play a critical role in the theory of cardinality (set size).
Click http://web.archive.org/web/20160307063048/http://ndp.jct.ac.il/tutorials/discrete/node42.html link to open resource.