Page 515

**Test Yourself**

1. If $R$ is a relation from $A$ to $B$, $x \in A$, and $y \in B$, the notation
   $x R y$ means that ____.

$x$ is related to $y$ by $R$

2. If $R$ is a relation from $A$ to $B$, $x \in A$, and $y \in B$, the notation
   $x \cancel{R} y$ means that ____.

$x$ is not related to $y$ by $R$.

3. If $R$ is a relation from $A$ to $B$, $x \in A$, and $y \in B$, the notation
   $(y, x) \in R^{-1}$ if, and only if, ____.

$$ (x, y) \in R $$

4. A relation on a set $A$ is a relation from ____ to ____.

$A$; $A$

5. If $R$ is a relation on a set $A$, the directed graph of $R$ has an arrow
   from $x$ to $y$ if, and only if, ____.

$x$ is related to $y$ by $R$

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Page 526

**Test Yourself**

1. For a relation $R$ on a set $A$ to be reflexive means that ____.

$\forall x \in A, x R x$

2. For a relation $R$ on a set $A$ to be symmetric means that ____.

$\forall x, y \in A, x R y \to y R x$

3. For a relation $R$ on a set $A$ to be transitive means that ____.

$\forall x, y, z \in A, (x R y \wedge y R z) \to x R z$

4. To show that a relation $R$ on an infinite set $A$ is reflexive, you suppose
   that ____ and you show that ____.

$x \in A$; $x R x$

5. To show that a relation $R$ on an infinite set $A$ is symmetric, you suppose
   that ____ and you show that ____.

$\forall x, y \in A, x R y$; $y R x$

6. To show that a relation $R$ on an infinite set $A$ is transitive, you suppose
   that ____ and you show that ____.

$\forall x, y, z \in A, x R y \wedge y R z$; $x R z$

7. To show that a relation $R$ on a set $A$ is not reflexive, you ____.

$\exists x \in A, x \cancel{R} x$

8. To show that a relation $R$ on a set $A$ is not symmetric, you ____.

$\exists x, y \in A, x R y \to y \cancel{R} x$

9. To show that a relation $R$ on a set $A$ is not transitive, you ____.

$\exists x, y, z \in A, (x R y \wedge y R z) \to x \cancel{R} z$

10. Given a relation $R$ on a set $A$, the transitive closure of $R$ is the
    relation $R^t$ on $A$ that satisfies the following three properties: ____,
    ____, and ____.

$R^t$ is transitive; $R \subseteq R^t$; if $S$ is any other transitive relation
that contains $R$, then $R^t \subseteq S$

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Page 543

**Test Yourself**

1. For a relation on a set to be an equivalence relation, it must be ____.

reflexive, symmetric, and transitive

2. The notation $m \equiv n (\mod d)$ is read "____" and means that ____.

$m$ is congruient to $n$ modulo $d$; $d$ divides $m - n$

3. Given an equivalence relation $R$ on a set $A$ and given an element $a$ in
   $A$, the equivalence class of $a$ is denoted ____ and is defined to be ____.

$[a]$; the set of all elements $x \in A$ such that $x R a$

4. If $A$ is a set, $R$ is an equivalence relation on $A$, and $a$ and $b$ are
   elements of $A$, then either $[a] = [b]$ or ____.

$[a] \cap [b] = \emptyset$

5. If $A$ is a set and $R$ is an equivalence relation on $A$, then the distinct
   equivalence classes of $R$ form ____.

a partition of $A$

6. Let $A = \mathbb{Z} \times (\mathbb{Z} - \{0\})$, and define a relation $R$
   on $A$ by specifying that for every $(a, b)$ and $(c, d)$ in $A$,
   $(a, b) R (c, d)$ if, and only if, $ad = bc$. Then there is exactly one
   equivalence class of $R$ for each ____.

rational number

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Page 566

**Test Yourself**

1. When letters of the alphabet are encrypted using the Caesar cipher, the
   encrypted version of the letter is ____.

2. If $a$, $b$, and $n$ are integers with $n > 1$, all of the following are
   different ways to express the fact that $n | (a - b)$: ____, ____, ____,
   ____.

3. If $a$, $b$, $c$, $d$, $m$, and $n$ are integers with $n > 1$ and if
   $a \equiv c(\mod n)$ and $b \equiv d(\mod n)$, then $a + b \equiv$ ____,
   $a - b \equiv$ ____, $ab \equiv$ ____, and $a^m \equiv$ ____

4. If $a$, $n$, and $k$ are positive integers with $n > 1$, an efficient way to
   compute $a^k(\mod n)$ is to write $k$ as a ____ and use the facts about
   computing products and powers modulo $n$.

5. To express a greatest common divisor of two integers as a linear combination
   of the integers, use the extended version of the ____ algorithm.

6. To find an inverse for a positive integer $a$ modulo an integer $n$ with
   $n > 1$, you express the number $1$ as ____.

7. To encrypt a message $M$ using RSA cryptography with public key $pq$ and $e$,
   you use the formula ____, and to decrypt a message $C$, you use the formula
   ____, where ____.

8. Euclid's lemma says that for all integers $a$, $b$, and $c$ if
   $\text{gcd}(a, c) = 1$ and $a | bc$, then ____.

9. Fermat's little theorem says that if $p$ is any prime number and $a$ is any
   integer such that $p | a$, then ____.

10. The crux of the proof that the RSA cipher works is that if (1) $p$ and $q$
    are distinct large prime numbers, (2) $M < pq$, (3) $M$ is relatively prime
    to $pq$, (4) $e$ is relatively prime to $(p - 1)(q - 1)$, and (5) $d$ is a
    positive inverse for $e$ modulo $(p - 1)(q - 1)$, then $M =$ ____.