How does one measure the size of a set, and is there a way of saying one set is bigger than another, even when they are both infinite? What "numbers" do measure size of infinite sets?
Size of A Set
One Set is Bigger than One Another even if they are both infinite!
Normally, people believe that it is easy to recognize the size of finite set rather than infinite set because we can count the number of elements in finite sizes. For instance, if a class holds 40 people and only 35 people attend the class, thus, we know that the number of seats is larger than the number of people since 40 is greater than 35.
However, you will surprise to know that it is no different if we want to compare the size of infinite sets too! Using the same example above, we can still know that the number of seats is larger than the number of people even if we don’t know the number (how many) of them! How? We can ask everyone to sit down and if there are number of seats left over, so we can conclude that there are more seats than people. This technique is used by Georg Cantor, a great Mathematician to compare the size of sets.
To explain this mathematically, let assume that X and Y are sets. Each element of X is matched with one and only one element of Y. This process is called as a one-to-one correspondence between sets X and Y. If we can construct the one-to-one correspondence between them, then we can say X and Y have the same size (X and Y have the same cardinality).
However, IF every trial to construct the one-to-one correspondence leaves X with elements that are not matched with elements of Y,
THEN we can say that X is larger than Y. (The cardinality of X is larger than the cardinality of Y)
