Figure 5

3. Let S = {1, 2, 3, 4, 5} and T = {a, b, c, d}. For each question below: if the answer is YES, give an example; if the answer if NO, explain briefly.

(a) Are there any one-to-one functions from S into T?

(b) Are there any one-to-one functions from T into S?

(c) Are there any functions mapping S onto T?

(d) Are there any functions mapping T onto S?

(e) Are there any one-to-one correspondences between S and T?

4. Let S = {1, 2, 3, 4, 5} and consider the following functions from S into S: l_s(n) = n, f(n) = 6 - n, g(n) = max{3, n}, h{n) = max{l, n - 1}.

(a) Write each of these functions as a set of ordered pairs; i.c., list the ele­ments in their graphs.

(b) Sketch a graph of each of these functions.

(c) Which of these functions are both one-to-one and onto?

5. The rule f((m, n)) = 2^m3ⁿ defines a one-to-one function from N  N into M. Note. When functions are defined on ordered pairs, it is customary to omit one set of parentheses. Thus we will write f((m, n)) = 2^m3ⁿ.

(a) Calculate f(m, n) for five different elements (m, n) in M  M.

(b) Explain why f is one-to-one.

(c) Does f map M  N onto N? Explain.

(d) Show that g(m, n) = 2^m4ⁿ defines a function on M  M that is not one-to-one.

6. Consider the following functions from N into N: l_N(n) = n, f(n) = 3n, g(n) = n + (-1)ⁿ, h(n} = min{n, 100}, k(n) = max{0, n - 5}.

(a) Which of these functions are one-to-one?

(b) Which of these functions map N onto N?

7. Let A and B be nonempty sets. The projection map proj picks the first element from each pair in A  B, i.e., proj: A  B  A is defined by proj(a, b) = a. [Recall the convention about functions on ordered pairs mentioned in Exer­cise 5.]

(a) Does this function map A  B onto A? Justify.

(b) Is proj one-to-one? What if B has only one element?

8. Let  = {a, b, c} and let ^ be the set of all words w using letters from ;

see Example 3(b). Define L(w) = length(w) for all w  ^*.

(a) Calculate L(w) for the words w₁ = cab, w₂ = ababac and w₃ = .

(b) Is L a one-to-one function? Explain.

(c) The function L maps ^* into M. Does L map ^* onto N? Explain.

(d) Find all words w such that L(w) = 2.

9. For n  Z, let f(n) = [(- 1)ⁿ + 1]. The function f is the characteristic func­tion for some subset of Z. Which subset?

9. In Example 5(b), we compared the functions and log . Show that these functions take the same value for x = 10,000.

11. We define three functions mapping R into R as follows: f(x) = x³ - 4x, g{x) = l/(x² + +1), h(x) = x⁴. Find

(a) f ° g ° h (b) f ° h ° g (c) h ° g ° f

(d) f ° f (e) g ° g (f) h ° g

(g) g ° h

12. Show that if f: S  T and g: T  U are one-to-one, then g ° f is one-to-one.

13. Prove that the composition of functions is associative.

14. Several important functions can be found on hand-held calculators. Why isn't the identity function, i.e., the function l_R where l_R (x) = x for all x  R, among them?

15. Consider the functions f and g mapping Z into Z, where f(n) = n - 1 for n  Z and g is the characteristic function _E of E = {n  Z :n is even}.

(a) Calculate (g o f)(5), (g o f)(4), (f o g)(7) and (f o g)(8).

(b) Calculate (f o f)(11), (f o f)(12), (g o g)(11) and (g o g)(l2).

(c) Determine the functions g o f and f o f.

(d) Show that g o g = g o f and that f o g is the negative of g o f.