discrete_mathematics_with_applications/chapter_8/test_yourself.md

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 = ____.