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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
\(\powset{A}_k\text{.}\)
