Example:
Suppose you are standing in an empty classroom, with a lot of students
waiting to get in. How can you know whether there are enough chairs for
everyone? Use the mathematical definition of cardinality to determine the
answer.
We can not count the students, since they are moving around too much.
Therefore, we set up a function f that associates each chair with a
student by simply asking each student to find a chair and sit down. If all
chairs are taken, and no students are left standing, then what does this mean
for our function f ?
- the domain of f is the space of all students
- the range of f is the space of all chairs
- the function f is one-to-one, because: if f(a) = f(b),
then student a and student b are occupying the same chair.
This can not happen unless student a and student b is the same.
Hence, f is injective.
- the function f is onto, because: the range is the space of
all chairs. Since all chairs are occupied, there is a student associated with
each chair. Hence, f is surjective.
Therefore, the function f is a bijection between the domain and the
range, and by definition of cardinality the number of students matches the
number of chairs, i.e. both sets have the same cardinality.
On the other hand, if the two sets did not contain the same number of elements,
the following could happen:
- if f was one-to-one, but not onto, then there would be empty
chairs. Hence, cardinality of the students is less than the cardinality of
the chairs.
- if f was onto, but not one-to-one, then all chairs are taken,
but some chairs hold more than one student. Hence, cardinality of the students
is greater than the cardinality of the chairs.
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