12.6: Exercises

Exercise \(\PageIndex{1}\)

Prove: If \(B\) is finite and \(A \subseteq B\text{,}\) then \(A\) is finite and \(\vert A \vert \le \vert B \vert\text{.}\)

Exercise \(\PageIndex{2}\)

Suppose that \(A\text{,}\) \(B\text{,}\) and \(C\) are finite subsets of a universal set \(U\text{.}\)

1. Prove: If \(A\) and \(B\) are disjoint, then \(\vert A \sqcup B \vert = \vert A \vert + \vert B \vert\text{.}\)
2. Prove: \(\vert A \cup B \vert = \vert A \vert + \vert B \vert - \vert A \cap B \vert\text{.}\)

Hint.

See Exercise 9.9.5, and use the equality from Task a.

3. Determine a similar formula for \(\vert A \cup B \cup C \text{.}\)

Hint.

Draw a Venn diagram first.

Exercise \(\PageIndex{3}\)

Use induction to prove directly that if \(\vert A \vert = n\) then \(\vert \mathscr{P}(A) \vert = 2^n\text{.}\) Use Worked Example 12.2.1 as a model for your proof of the induction step.

Exercise \(\PageIndex{4}\)

Prove: If \(\vert A \vert = \infty\) and \(A \subseteq B\text{,}\) then \(\vert B \vert = \infty\text{.}\)

Exercise \(\PageIndex{5}\)

Prove Fact 12.3.2.

Exercise \(\PageIndex{6}\)

Combine Example 12.3.3 and Example 12.3.10 to verify that the unit interval \((0,1)\) and \(\mathbb{R}\) have the same size.

Hint.

First map the punctured circle \(\hat{S}\) onto some open interval in the \(x\)-axis by “unrolling” \(\hat{S}\text{.}\)

Exercise \(\PageIndex{7}\)

Use Example 12.3.3 and the function \(f(x) = \tan x\) to prove that the interval \((-\pi/2,\pi/2)\) and \(\mathbb{R}\) 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

Exercise \(\PageIndex{8}\)

Prove that if \(A\) and \(B\) have the same size, then so do \(\mathscr{P}(A)\) and \(\mathscr{P}(B)\text{.}\)

Hint.

See Exercise 10.7.19.

Exercise \(\PageIndex{9}\)

Suppose \(A\) is a set with \(\vert A \vert = n\text{.}\) Then we can enumerate its elements as \(A = \{a_1,a_2,\ldots ,a_n\}\text{.}\)

1. 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.

1. Explicitly describe a function \(f: \mathscr{P}(A) \rightarrow \Sigma^\ast_n\) 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.
2. Prove that your function \(f\) is a bijection.

Hint.

When determining the input-output rule for your function \(f: \mathscr{P}(A) \rightarrow \Sigma^\ast_n\text{,}\) 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.

2. Use Task a to determine the cardinality of \(\mathscr{P}(A)\text{.}\) Explain.

Hint

See Note 1.3.1.

3. Suppose \(k\) is some fixed (but unknown) integer, with \(0 \le k \le n\text{.}\) Let \(\mathscr{P}(A)_k\) represent the subset of \(\mathscr{P}(A)\) 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 \(\mathscr{P}(A)_k\text{.}\)

Hint.

Consider how restricting the domain might help.