Section 6: Problem 1 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Make a list of all the injective maps
Show that none is bijective. (This constitutes a direct proof that a set
of cardinality three does not have cardinality four.)
(b) How many injective maps
are there? (You can see why one would not wish to try to prove directly that there is no bijective correspondence between these sets.)
(a) An injective correspondence will have its image set consisting of three elements of
. We can exclude one of the four elements from the set
, and for each case there are
ways to define a bijective correspondence of
with the remaining set of 3 elements. You can explicitly write down these
injective correspondences.
(b) Similarly, we can exclude
elements from
(
combinations), and define a bijection between
and the remaining subset of
consisting of
elements (
combinations). The total number of injective functions is
combinations. If you spend
seconds on average to write down each bijective correspondence, it will take you
days to complete the list.