🚧 Setup for 8.1

chapter_8/exercises.md

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+**Exercise Set 8.1**
+
+1. As in Example 8.1.2, the **congruence modulo $2$** relation $E$ is defined
+   from $\mathbb{Z}$ to $\mathbb{Z}$ as follows: For every ordered pair
+   $(m, n) \in \mathbb{Z} \times \mathbb{Z}$,
+
+$$ m E n \Leftrightarrow m - n \text{ is even} $$
+
+a. Is $0 E 0$? Is $5 E 2$? Is $(6, 6) \in E$? Is $(-1, 7) \in E$?
+
+b. Prove that for any even integer $n$, $n E 0$.
+
+2. Prove that for all integers $m$ and $n$, $m - n$ is even if, and only if,
+   both $m$ and $n$ are even or both $m$ and $n$ are odd.
+
+3. The **congruence modulo $3$** relation, $T$, is defined from $\mathbb{Z}$ to
+   $\mathbb{Z}$ as follows: For all integers $m$ and $n$,
+
+$$ m T n \Leftrightarrow 3 | (m - n) $$
+
+a. Is $10 T 1$? Is $1 T 10$? Is $(2, 2) \in T$? Is $(8, 1) \in T$?
+
+b. List five integers $n$ such that $n T 0$.
+
+c. List five integers $n$ such that $n T 1$.
+
+d. List five integers $n$ such that $n T 2$.
+
+e. Make and prove a conjecture about which integers are related by $T$ to $0$,
+which integers are related to $T$ to $1$, and which integers are related to $T$
+to $2$.
+
+4. Define a relation $P$ on $\mathbb{Z}$ as follows: For every ordered pair
+   $(m, n) \in \mathbb{Z} \times \mathbb{Z}$,
+
+$$ m P n \Leftrightarrow m \text{ and } n \text{ have a common prime factor} $$
+
+a. Is $15 P 25$?
+
+b. Is $22 P 27$?
+
+c. Is $0 P 5$?
+
+d. Is $8 P 8$?
+
+5. Let $X = \{a, b, c\}$. Recall that $\mathscr{P}(X)$ as follows: For all sets
+   $A$ and $B$ in $\mathscr{P}(X)$,
+
+$$ A \mathbf{J} B \Leftrightarrow A \cap B \neq \emptyset $$
+
+a. Is $\{a\} \mathbf{J} \{c\}$?
+
+b. Is $\{a, b\} \mathbf{J} \{b, c\}$?
+
+c. Is $\{a, b} \mathbf{J} \{a, b, c\}$?
+
+7. Define a relation $R$ on $\mathbb{Z}$ as follows: For all integers $m$ and
+   $n$,
+
+$$ m R n \Leftrightarrow 5 | (m^2 - n^2) $$
+
+a. Is $1 R (-9)$?
+
+b. Is $2 R 13$?
+
+c. Is $2 R (-8)$?
+
+d. Is $(-8) R 2$?
+
+8. Let $A$ be the set of all strings of _a_'s and _b_'s of length $4$. Define a
+   relation $R$ on $A$ as follows: For every $s, t \in A$,
+
+$$ s R t \Leftrightarrow s \text{ has the same first two characters as } t $$
+
+a. Is _abaa_ $R$ _abba_?
+
+b. Is _aabb_ $R$ _bbaa_?
+
+c. Is _aaaa_ $R$ _aaab_?
+
+d. Is _baaa_ $R$ _abaa_?
+
+9. Let $A$ be the set of all strings of 0's, 1's, and 2's of length $4$. Define
+   a relation $R$ on $A$ as follows: For every $s, t \in A$,
+
+$$ s R t \Leftrightarrow \text{ the same of the characters in } s \text{ equals the sum of the characters in } t $$
+
+a. Is 0121 $R$ 2200?
+
+b. Is 1011 $R$ 2101?
+
+c. Is 2212 $R$ 2121?
+
+d. Is 1220 $R$ 2111?
+
+10. Let $A = \{3, 4, 5\}$ and $B = \{4, 5, 6\}$ and let $R$ be the "less than"
+    relation. That is, for every ordered pair $(x, y) \in A \times B$,
+
+$$ x R y \Leftrightarrow x < y $$
+
+State explicitly which ordered pairs are in $R$ and $R^{-1}$.
+
+11. Let $A = \{3, 4, 5\}$ and $B = \{4, 5, 6\}$ and let $S$ be the "divides"
+    relation. That is, for every ordered pair $(x, y) \in A \times B$,
+
+$$ x S y \Leftrightarrow x | y $$
+
+State explicitly which ordered pairs are in $S$ and $S^{-1}$.
+
+12.
+
+a. Suppose a function $F: X \to Y$ is one-to-one but not onto. Is $F^{-1}$ (the
+inverse relation for $F$) a function? Explain your answer.
+
+b. Suppose a function $F: X \to Y$ is onto but not one-to-one. Is $F^{-1}$ (the
+inverse relation for $F$) a function? Explain your answer.
+
+Draw the directed graphs of the relations defined in 13-18.
+
+13. Define a relation $R$ on $A = \{0, 1, 2, 3\}$ by
+    $R = \{(0, 0), (1, 2), (2, 2)\}$.
+
+14. Define a relation $S$ on $B = \{a, b, c, d\}$ by
+    $S = \{(a, b), (a, c), (b, c), (d, d)\}$.
+
+15. Let $A = \{2, 3, 4, 5, 6, 7, 8\}$ and define a relation $R$ on $A$ as
+    follows: For every $x, y \in A$,
+
+$$ x R y \Leftrightarrow x | y $$
+
+16. Let $A = \{5, 6, 7, 8, 9, 10\}$ and define a relation $S$ on $A$ as follows:
+    For every $x, y \in A$,
+
+$$ x S y \Leftrightarrow 2 | (x - y) $$
+
+17. Let $A = \{2, 3, 4, 5, 6, 7, 8\}$ and define a relation $T$ on $A$ as
+    follows: For every $x, y \in A$,
+
+$$ x T y \Leftrightarrow 3 | (x - y) $$
+
+18. Let $A = \{0, 1, 3, 4, 5, 6\}$ and define a relation $V$ on $A$ as follows:
+    For every $x, y \in A$,
+
+$$ x V y \Leftrightarrow 5 | (x^2 - y^2) $$
+
+Exercises 19-20 refer to unions and intersections of relations. Since relations
+are subsets of Cartesian products, their unions and intersections can be
+calculated as for any subsets. Given two relations $R$ and $S$ from $A$ to $B$,
+
+$$ R \cup S = \{(x, y) \in A \times B | (x, y) \in R \text{ or } (x, y) \in S\} $$
+
+$$ R \cap S = \{(x, y) \in A \times B | (x, y) \in R \text{ and } (x, y) \in S\} $$
+
+19. Let $A = \{2, 4\}$ and $B = \{6, 8, 10\}$ and define relations $R$ and $S$
+    from $A$ to $B$ as follows: For every $(x, y) \in A \times B$,
+
+$$ x R y \Leftrightarrow x | y \text{ and } x S y \Leftrightarrow y - 4 = x $$
+
+State explicitly which ordered pairs are in $A \times B$, $R$, $S$, $R \cup S$,
+and $R \cap S$.
+
+20. Let $A = \{-1, 1, 2, 4\}$ and $B = \{1, 2\}$ and define relations $R$ and
+    $S$ from $A$ to $B$ as follows: For every $(x, y) \in A \times B$,
+
+$$ x R y \Leftrightarrow |x| = |y| \text{ and } x S y \Leftrightarrow x - y \text{ is even} $$
+
+State explicitly which ordered pairs are in $A \times B$, $R$, $S$, $R \cup S$,
+and $R \cap S$.
+
+21. Define relations $R$ and $S$ on $\mathbb{R}$ as follows:
+
+$$ R = \{(x, y) \in \mathbb{R} \times \mathbb{R} | x < y\} \text{ and } S = \{(x, y) \in \mathbb{R} \times \mathbb{R} | x = y\}$$
+
+That is, $R$ is the "less than" relation and $S$ is the "equals" relation on
+$\mathbb{R}$. Graph $R$, $S$, $R \cup S$, and $R \cap S$ in the Cartesian plane.
+
+22. Define relations $R$ and $S$ on $\mathbb{R}$ as follows:
+
+$$ R = \{(x, y) \in \mathbb{R} \times \mathbb{R} | x^2 + y^2 = 4\} \text{ and } S = \{(x, y) \in \mathbb{R} \times \mathbb{R} | x = y\} $$
+
+Graph $R$, $S$, $R \cup S$, and $R \cap S$ in the Cartesian plane.
+
+23. Define relations $R$ and $S$ on $\mathbb{R}$ as follows:
+
+$$ R = \{(x, y) \in \mathbb{R} \times \mathbb{R} | y = |x|\} \text{ and } S = \{(x, y) \in \mathbb{R} \times \mathbb{R} | y = 1\} $$
+
+Graph $R$, $S$, $R \cup S$, and $R \cap S$ in the Cartesian plane.
+
+24. In Example 8.1.7 consider the query SELECT Patient_ID#, Name FROM S WHERE
+    Primary_Diagnosis = X. The response query is the projection onto the first
+    two coordinates of the intersection of the database with the set
+    $A_1 \times A_2 \times A_3 \times \{X\}$.
+
+a. Find the result of the query SELECT Patient_ID#, Name FROM S WHERE
+Primary_Diagnosis = pneumonia.
+
+b. Find the result of the query SELECT Patient_ID#, Name FROM S WHERE
+Primary_Diagnosis = appendicitis.