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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.