1. ## cardinality proof

How can I prove the following:

if the cardinality of A1 = the cardinality of A2, the cardinality of B1 = the cardinality of B2, B1 is a subset of A1, and B2 is a subset of A2,
then the cardinality of (A1 / B1) = the cardinality of (A2 / B2) where / is the set difference operation

I know there exists a bijective function f: A1 to A2 and a bijective function g: B1 to B2. I need to find a bijective function h: (A1 / B1) to (A2 / B2). Can someone fill in the details?

2. Consider this example: $\displaystyle B_1 = \left\{ { \cdots ,-6, - 4, - 2} \right\} \subseteq \mathbb{Z}^ - \;\& \;A_1 = \left\{ {5,6, \cdots } \right\} \subseteq \mathbb{Z}^ +$.
It is clear that $\displaystyle \mathbb{Z}^ - \leftrightarrow \mathbb{Z}^ +$.
Define $\displaystyle \Phi :A_1 \leftrightarrow B_1 ,\;\;\Phi (a) = - 2\left( {a - 4} \right)$. Prove that $\displaystyle \Phi$ is a bijection.
Now consider $\displaystyle \left( {\mathbb{Z}^ - \backslash B_1 } \right)\;\& \;\left( {\mathbb{Z}^ + \backslash A_1 } \right)$.

3. Originally Posted by PvtBillPilgrim
How can I prove the following:

if the cardinality of A1 = the cardinality of A2, the cardinality of B1 = the cardinality of B2, B1 is a subset of A1, and B2 is a subset of A2,
then the cardinality of (A1 / B1) = the cardinality of (A2 / B2) where / is the set difference operation

I know there exists a bijective function f: A1 to A2 and a bijective function g: B1 to B2. I need to find a bijective function h: (A1 / B1) to (A2 / B2). Can someone fill in the details?