Homework1

Assignment 1

Sets

1. Given sets A and B, show that $A \cap B$ is the largest common subset of A and B, in the sense that it contains every such common subset.

Note:  Typically for statements involving largest or smallest, a proof by contradiction works quite well.

2. Given sets A and B, prove if $A \cap X = B \cap X$ and $A \cup X = B \cup X$ for some set X, then A = B

Hint:  $A = A \cap (A \cup X)$

Functions

1. Given functions $f:A \rightarrow B$ and $g:B \rightarrow C$, prove that if $g \circ f$ is one-to-one (injective), then f is one-to-one.

Note: The function g need not be injective nor surjective.