Part 1: Functions:
1. (4 points) For each of the following relations, determine if it is a function. Justify your answer.
(a) f(x) = 1
x3 from Z to R
(b) f(x) = px from Z+ to R

2. (4 points) For each of the following functions, give the domain, codomain, and range.
(a) f(x) = x%5.
(b) The function that, given a bit string, returns the length of the bit string.

3. (4 points) Evaluate each of the following:
(a) d 3:5e
(b) d 2:2  d 5
2 ee
(c) b2:99c
(d) b 4
3 + d 1
3 ec

4. (6 points) For each of the following functions determine if it is injective. Justify your answer.
(a) f(n) = n5 from Z to Z
(b) f(n) = jnj + 1 from Z to Z+
(c) f(n) = n + 3 from Z to Z

5. (6 points) Determine if each of the following functions from question 4 are surjective. Justify your
(a) f(n) = n5 from Z to Z
(b) f(n) = jnj + 1 from Z to Z+
(c) f(n) = n + 3 from from Z to Z

6. (6 points) Which of the functions from questions 4 and 5 are bijective? Brie y justify your answer.
(a) f(n) = n5 from Z to Z
(b) f(n) = jnj + 1 from Z to Z+
(c) f(n) = n + 3 from from Z to Z

7. (4 points) For each of the following, give an example of a function from Z to Z that satis es the
speci ed properties. You may not use the functions from the previous three problems.
(a) One-to-one but not onto
(b) Onto but not one-to-one

8. (4 points) Let C be the set of CS Faculty. Consider the following functions whose domains are C.
Answer the following for each function: (i) Under what conditions is the function one-to-one? (ii)
Under what conditions is the function onto?
(a) The function f from C to O that assigns faculty to oces, where O is the set of oces on the
7th oor of Gould Simpson.
(b) The function g from C to S that assigns faculty to the courses they will teach this semester,
where S is the set of courses o ered by the CS department. Assume each faculty teaches
exactly one course (otherwise it’s not a function!)

9. (4 points) Prove or disprove the following conjecture: d3xe = 3  dxe.
10. (8 points) Prove the following conjecture: bxc + byc  bx + yc. (Hint: Use the fact that, for any
real number x, x = m + d where m 2 Z, d 2 R with m  x < m + 1 and 0  d < 1). EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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