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Exercises 12.6 Exercises
1. Prove: If
\(B \) is finite and
\(A \subseteq B \text{,}\) then
\(A \) is finite and
\(\card{A} \le \card{B} \text{.}\)
2.
Suppose that
\(A \text{,}\) \(B \text{,}\) and
\(C \) are finite subsets of a universal set
\(U \text{.}\)
(a)
Prove: If
\(A \) and
\(B \) are disjoint, then
\(\card{A \sqcup B} = \card{A} + \card{B} \text{.}\)
(b)
Prove: \(\card{A \cup B} = \card{A} + \card{B} - \card{A \cap B} \text{.}\)
(c) Determine a similar formula for
\(\card{A \cup B \cup C} \text{.}\)
Hint .
Draw a Venn diagram first.
3. Use induction to prove directly that if
\(\card{A} = n \) then
\(\card{\powset{A}} = 2^n \text{.}\) Use
Worked Example 12.2.7 as a model for your proof of the induction step.
4. Prove: If
\(\card{A} = \infty \) and
\(A \subseteq B \text{,}\) then
\(\card{B} = \infty \text{.}\)
5. 6.
Hint .
First map the punctured circle
\(\hat{S} \) onto some open interval in the
\(x \) -axis by “unrolling”
\(\hat{S} \text{.}\)
7. Use
Example 12.3.7 and the function
\(f(x) = \tan x \) to prove that the interval
\((-\pi/2,\pi/2) \) and
\(\R \) have the same size.
Hint .
The function
\(f(x) = \tan x \) is not one-to-one, but it becomes one-to-one if you restrict its domain to an appropriate interval
8. Prove that if
\(A \) and
\(B \) have the same size, then so do
\(\powset{A} \) and
\(\powset{B} \text{.}\)
9.
Suppose
\(A \) is a set with
\(\card{A} = n \text{.}\) Then we can enumerate its elements as
\(A = \{a_1,a_2,\dotsc,a_n\} \text{.}\)
(a) Construct a bijection from the power set of
\(A \) to the set of words in the alphabet
\(\Sigma = \{T,F\} \) of length
\(n \text{.}\)
Note that there are two tasks required here.
Explicitly describe a function
\(\funcdef{f}{\powset{A}}{\words{\Sigma}_n} \) by describing the
input-output rule : give a detailed description of how, given a subset
\(B \subseteq A \text{,}\) the word
\(f(B) \) should be produced.
Prove that your function \(f \) is a bijection.
Hint .
When determining the input-output rule for your function
\(\funcdef{f}{\powset{A}}{\words{\Sigma}_n} \text{,}\) think of how one might construct an arbitrary subset of
\(A \text{,}\) and then relate that process to a sequence of answers to
\(n \) true/false questions.
(b) Use
Task a to determine the cardinality of
\(\powset{A} \text{.}\) Explain.
(c) Suppose
\(k \) is some fixed (but unknown) integer, with
\(0 \le k \le n \text{.}\) Let
\(\powset{A}_k \) represent the subset of
\(\powset{A} \) consisting of all subsets of
\(A \) that have exactly
\(k \) elements. Describe how your bijection from
Task a , could be used to count the elements of
\(\powset{A}_k \text{.}\)
