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\}

---

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, _____.

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, _____.

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

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

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) 

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 _____.

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 _____.

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

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

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

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 _____.