Note that it is dangerous to redefine symbols like ≤ and ≥! People expect that these relations work the same way they do for numbers, but it is not a-priori obvious that they do. For example, you might expect that
- |A| ≤ |A|
- if |A| ≤ |B| and |B| ≤ |C| then |A| ≤ |C|
- if |A| ≤ |B| then |B| ≥ |A|
- if |A| ≤ |B| and |A| ≥ |B| then |A| = |B|
- if A ⊆ B then |A| ≤ |B|
Luckily, these are all true (most will be proved on the homework), but it's important to note that they need to be proved!