DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

24. If f is a bijection from X to Y and g is a bijection from Y to Z, must
the function composition g ◦ f be a bijection from X to Z? Explain.
25. If you reverse all the arrows in the digraph of a bijection f with domain
S and co-domain T , you get the digraph of another function g. Why
is g a function from T to S? Why is g a bijection? What is f(g(x))?
What is g(f(x))?
The digraphs marked (a), (b), and (e) in Figure 1.3 are digraphs of
bijections. Your description in Problem 23 illustrates another fundamental
principle of combinatorial mathematics:
The Bijection Principle
Two sets have the same size if and only if there is a bijection between the
sets.
It is surprising how this innocent-sounding principle frequently provides an
insight into some otherwise very complicated counting arguments.
A binary representation of a positive integer m is an ordered list
a1a2 . . . ak of zeros and ones such that
m = a12 k−1 + a22 k−2 + · · · + ak2 0 .
Our definition allows “leading zeros”: for instance, both 011 and 11 represent
the number 3. Often such an ordered list of k zeros and ones is called a
binary k-string, and each of its digits is called a bit, which is shorthand
for binary digit. In the above example, 011 is a binary 3-string representation
of 3, while 11 is a binary 2-string representation of 3.
•26. Let n be a fixed positive integer. For this problem, let S be the set of
all binary n-string representations of numbers between 0 and 2 n − 1,
and let T be the set of all subsets of [n]. Note that the empty set ∅ is
a subset of every set.
(a) For n = 2, write out the sets S and T and then describe a natural
bijection from S to T where in your statement of the bijection,
0 and 1 play the roles of “does not belong to” and “belongs”,
respectively.
(b) Using the strategy from part (a), describe a bijection from S to T
for general n. Explain why your map is a bijection from S to T .
(c) Explain how part (b) enables you to find the number of subsets
of [n].
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