discrete_mathematics_with_applications/chapter_7/test_yourself.md

Page 458

Test Yourself

1. Given a function f from a set X to a set Y, f(x) is _____.

the unique output element in Y that is related to x by f.

2. Given a function f from a set X to a set Y, if f(x) = y then y is called _____ or _____ or _____.

the value of f at x; the image of x under f; the output of f for the input x

3. Given a function f from a set X to a set Y, the range of f (or the image of X under f) is _____.

the set of all y in Y such that f(x) = y

4. Given a function f from a set X to Y, if f(x) = y then x is called _____ or _____.

an inverse image of y under f; a preimage of y

5. Given a function f from a set X to a set Y, if y \in Y then f^{-1}(y) = _____ and is called _____.

\{x \in X | f(x) = y\}; the inverse image of y

6. Given functions f and g from a set X to a set Y, f = g if, and only if, _____.

f(x) = g(x) for every x \in X

7. Given positive real numbers x and b with b \neq 1, \log_b(x) = _____.

the exponent to which b must be raised to obtain x.

8. Given a function f from a set X to a set Y and a subset A of X, f(A) = _____.

\{y \in Y | y = f(x) \text{ for some } x \in A\}

9. Given a function f from a set X to a set Y and a subset C of Y, f^{-1}(C) = _____.

\{x \in X | f(x) \in C\}

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Page 480

Test Yourself

1. If F is a function from a set X to a set Y, then F is one-to-one if, and only if, _____.

for all x_1 and x_2 in X, if F(x_1) = F(x_2) then x_1 = x_2

2. If F is a function from a set X to a set Y, then F is not one-to-one if, and only if, _____.

for all x_1 and x_2 in X, if F(x_1) = F(x_2) then x_1 \neq x_2

3. If F is a function from a set X to a set Y, then F is onto if, and only if, _____.

for every element y in Y, there exists at least one element x in X such that f(x) = y

4. If F is a function from a set X to a set Y, then F is not onto if, and only if, _____.

for every element y in Y, there exists at least one element x in X such that f(x) \neq y

5. The following two statements are _____:

 \forall u, v \in U, \text{ if } H(u) = H(v) \text{ then } u = v 

 \forall u, v \in U, \text{ if } u \neq v \text{ then } H(u) \neq H(v) 

logically equivalent ways of expressing what it means for a function H to be one-to-one (The second is the contrapositive of the first.)

6. Given a function F: X \to Y where X is an infinite set, to prove that F is one-to-one, you suppose that _____ and then you show that _____.

x_1 and x_2 are any [particular but arbitrarily chosen] elements in X with the property that F(x_1) = F(x_2); x_1 = x_2

7. Given a function F: X \to Y where X is an infinite set, to prove that F is onto, you suppose that _____ and then you show that _____.

y is any [particular but arbitrarily chosen] element in Y; there exists at least one element x in X such that F(x) = y

8. Given a function F: X \to Y, to prove that F is not one-to-one, you _____.

show that there are concrete elements x_1 and x_2 in X with the property that F(x_1) = F(x_2) and x_1 \neq x_2

9. Given a function F: X \to Y, to prove that F is not onto, you _____.

show that there is a concrete element y in Y with the property that F(x) \neq y for any element x in X

10. A one-to-one correspondence from a set X to a st Y is a _____ that is _____.

function from X to Y; both one-to-one and onto

11. If F is a one-to-one correspondence from a set X to a set Y and y is in Y, then F^{-1}(y) is _____.

the unique element x in X such that F(x) = y (in other words, F^{-1}(y) is the unique preimage of y in X)

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Page 494

Test Yourself

1. If f is a function from X to Y', g is a function from Y \to Z, and Y' \subseteq Y, then g \circ f is a function from _____ to _____, and (g \circ f)(x) = _____ for every x in X.

X; Z, g(f(x))

2. If f is a function from X to Y and I_x and I_y are the identity functions from X to X and Y to Y, respectively, then f \circ I_x = _____ and I_y \circ f = _____.

f; f

3. If f is a one-to-one correspondence from X to Y, then f^{-1} \circ f = _____ and f \circ f^{-1} = _____.

I_X; I_Y

4. If f is a one-to-one function from X to Y and g is a one-to-one function from Y to Z, you prove that g \circ f is one-to-one by supposing that _____ and then showing that _____.

for some x_1, x_2 \in X, (g \circ f)(x_1) = (g \circ f)(x_2); x_1 = x_2

5. If f is an onto function from X to Y and g is an onto function from Y to Z, you prove that g \circ f is onto by supposing that _____ and then showing that _____.

for some z \in Z, there exists some x \in X, such that (g \circ f)(x) = z

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Page 507

Test Yourself

1. A set is finite if, and only if, _____.

it is the empty set or there is a one-to-one correspondence from \{1, 2, \dots n\} to it, for some positive integer n.

2. To prove that a set A has the same cardinality as a set B you must _____.

show that there is a function one-to-one correspondence from A to B.

3. The reflexive property of cardinality says that given any set A, _____.

A has the same cardinality as A.

4. The symmetric property of cardinality says that given any sets A and B, _____.

if A has the same cardinality as B, then B has the same cardinality as A.

5. The transitive property of cardinality says that given any sets A, B, and C, _____.

if A has the same cardinality as B, and if B has the same cardinality as C, then A has the same cardinality as C.

6. A set is called countably infinite if, and only if, _____.

it has the same cardinality as the set of all positive integers (\mathbb{Z}^+).

7. A set is called countable if, and only if, _____.

it is finite or countably infinite

8. In each of the following, fill in the blank with the word countable or the word uncountable.

a. The set of all integers is _____.

countable

b. The set of all rational numbers is _____.

countable

c. The set of all real numbers between 0 and 1 is _____.

uncountable

d. The set of all real numbers is _____.

uncountable

9. The Cantor diagonalization process is used to prove that _____.

the set of all real numbers between 0 and 1 is uncountable