9.9: Exercises
Prove each of the set operation rules in Proposition 9.4.1 . Use the provided proof of the first of DeMorgan's Laws as a model for your proofs.
Expressing relationships using the symbols of set theory.
In each of Exercises 2–4, you are given a collection of sets (and possibly some elements of those sets), a collection of symbols, and a collection of statements about those sets and their elements. Use the given symbols to express the given statements in symbolic language.
Note that there may be more than one correct answer for each statement.
Sets:
\begin{align*} A & = \text{ the set of all Augustana students, } \\ R & = \text{ the set of Augustana students who attend class regularly, } \\ S & = \text{ the set of Augustana students who study diligently, } \\ P & = \text{ the set of Augustana students who will pass all their courses. } \end{align*}
Symbols:
\begin{equation*} A, \quad R, \quad S, \quad P, \quad R^c, \quad S^c, \quad P^c, \quad \cap, \quad \cup, \quad =, \quad \ne, \quad \subseteq, \quad \subsetneqq, \quad \varnothing\text{.} \end{equation*}
Statements:
- All Augustana students who attend class regularly and study diligently will pass all their courses.
- Some Augustana students attend class regularly but do not study diligently.
- Some Augustana students who study diligently will still fail a course.
Recall that a square number is an integer which is equal to the square of some integer. (See the introduction preceding Exercise 6.12.17 in Section 6.12 .)
Sets:
\begin{align*} P & = \text{ the set of prime numbers, } \\ E & = \text{ the set of even numbers, } \\ S & = \text{ the set of square numbers. } \end{align*}
Symbols:
\begin{equation*} 2, \quad \N, \quad P, \quad E, \quad S, \quad \mathbb{N}^c, \quad P^c, \quad E^c, \quad S^c, \quad \in, \quad \cap, \quad \cup, \quad =, \quad \ne, \quad \subseteq, \quad \subsetneqq, \quad \varnothing, \quad \{\;\}. \end{equation*}
Statements:
- \(2\) is the only even, prime number.
- There exist odd square numbers.
- No prime number is square.
- No square number is prime.
- It is not true that every natural number is either even or prime.
Sets:
\begin{align*} \mathscr{F} & = \text{ the set of all functions in a single real variable, } \\ \mathscr{C} & = \text{ the set of continuous functions, } \\ \mathscr{D} & = \text{ the set of differentiable functions, } \\ \mathscr{P} & = \text{ nonnegative functions } \\ & = \{f(x) \vert f(x) \ge 0 \text{ for all } x \text{ in the domain of } f \}\text{.} \end{align*}
Elements:
\begin{align*} f_1(x) & = x^2 & f_2(x) & = \vert x \vert & f_3(x) & = \tan x \end{align*}
Symbols:
\begin{equation*} \mathscr{F}, \;\;\; \mathscr{C}, \;\;\; \mathscr{D}, \;\;\; \mathscr{P}, \;\;\; \mathscr{F}^c, \;\;\; \mathscr{C}^c, \;\;\; \mathscr{D}^c, \;\;\; \mathscr{P}^c, \;\;\; f_1(x), \;\;\; f_2(x), \;\;\; f_3(x), \;\;\; \in, \;\;\; \cap, \;\;\; \cup, \;\;\; =, \;\;\; \ne, \;\;\; \subseteq, \;\;\; \subsetneqq, \;\;\; \varnothing. \end{equation*}
Statements:
- The function \(f_1(x)\) is differentiable and nonnegative.
- The function \(f_2(x)\) is continuous and nonnegative, but not differentiable.
- The function \(f_3(x)\) is neither continuous nor nonnegative.
- Every differentiable function is continuous.
- Some continuous functions are not differentiable.
- Not every function is continuous.
Testing set equality
For each of Exercises 5–8, either formally prove the given equivalence of sets (using the Test for Set Equality) or demonstrate that it is false by providing a specific counterexample.
\(A = (A \setminus B) \sqcup (A \cap B)\)
\(A \setminus (A \setminus B) = B\)
\((A \times B) \cup (C \times D) = (A \cup C) \times (B \cup D)\)
\(A \times (B \setminus C) = (A \times B) \setminus (A \times C)\)
Suppose \(\Sigma\) is an alphabet. Prove that \(\Sigma^{\ast}\) is the disjoint union of the subsets
\begin{equation*} \Sigma^{\ast}_0, \Sigma^{\ast}_1, \Sigma^{\ast}_2, \dotsc, \Sigma^{\ast}_n, \dotsc \text{.} \end{equation*}
Write out the elements of each of the sets
\begin{align*} & \mathscr{P}( \varnothing ), & & \mathscr{P}( \, \mathscr{P}( \varnothing ) \, ), & & \mathscr{P}( \, \mathscr{P}( \, \mathscr{P}( \varnothing ) \, ) \, ), & & \mathscr{P}( \, \mathscr{P}( \, \mathscr{P}( \, \mathscr{P}( \varnothing ) \, ) \, ) \, )\text{.} \end{align*}
Make sure you have all the pairs of braces \(\{\;\}\) you should have.
Without computing it, make a conjecture about the number of elements in the set
\begin{equation*} \mathscr{P}( \, \mathscr{P}( \, \mathscr{P}( \, \mathscr{P}( \, \mathscr{P}( \varnothing ) \, ) \, ) \, ) \, ). \end{equation*}
Properties of power sets.
For each of Exercises 11–14, either formally prove the given statement about power sets or demonstrate that it is false by providing a specific counterexample.
\(\mathscr{P}(A \cup B) = \mathscr{P}(A) \cup \mathscr{P}(B)\)
\(\mathscr{P}(A \cap B) = \mathscr{P}(A) \cap \mathscr{P}(B)\)
If \(A \subseteq B\text{,}\) then \(\mathscr{P}(A) \subseteq \mathscr{P}(B)\text{.}\)
If \(A \subseteq B\text{,}\) then \(\mathscr{P}(B \setminus A) = \mathscr{P}(B) \setminus \mathscr{P}(A)\text{.}\)