2.5 Cardinality Flashcards
When do the sets A and B have the same cardinality?
|A| = |B|
The sets A and B have the same cardinality if and only if there is a one-to-one correspondence (bijection) from A to B.
Countable sets
Sets that are either finite or have the same cardinality as the set of positive integers.
How to express the cardinality of an infinite countable set
ℵ0 (aleph null)
How to show that the set of odd integers is countable
SCHRÖDER-BERNSTEIN THEOREM.
If A and B are sets with |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|.
If the function from A to B, and the function from B to A are one to one, then there is a one-to-one correspondence between A to B.
Show that (0,1) and (0,1] have the same cardinality
Use S-B theorem.
Find one-to-one function from (0,1) to (0,1] . f(x)= x
Then find a one-to-one function from (0,1] to (0,1). g(x) = x/2
Show that the set of all integers is countable