2.5 Cardinality

Question 1

When do the sets A and B have the same cardinality?
|A| = |B|

Answer 1

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.

Question 2

Countable sets

Answer 2

Sets that are either finite or have the same cardinality as the set of positive integers.

Question 3

How to express the cardinality of an infinite countable set

Answer 3

ℵ0 (aleph null)

Question 4

How to show that the set of odd integers is countable

Answer 4

Question 5

SCHRÖDER-BERNSTEIN THEOREM.

Answer 5

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.

Question 6

Show that (0,1) and (0,1] have the same cardinality

Answer 6

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

Question 7

Show that the set of all integers is countable

Answer 7