- This is a
**closed book and notes**exam. -
**No computer or calculators of any kind**are allowed. - Please make sure that
**your phone is turned off and out of reach**. - Please write your answers on the exam sheet, using the back if necessary.
- There are seven questions for 100 total points. Scale was A = 95, C = 75.
- Please work crefully and neatly and
**show your work**. - If you finish early, please read over your solutions and make sure that they are clear and correct. Use any extra time to improve their correctness and clarity.
- Partial credit will be given for partial solutions. Good luck!

Q1: 20 points Q2: 15 points Q3: 10 points Q4: 15 points Q5: 15 points Q6: 10 points Q7: 15 points Total: 100 points

**Question 1 (20):**Determine whether the following rules define functions from the given domain to co-domain, and if so, whether they are 1:1, onto, both, or neither. If you say that the function is 1:1 and onto, then also describe its inverse. Recall that Σ_{b}= {0, 1}.- (a) a:
**R**→**R**where a(x) = x^{3} - (b) b: Σ
_{b}× Σ_{b}→ Σ_{b}× Σ_{b}where b(x, y) = (x ⊕ y, y) - (c) c:
**R**→**R**where c(x) = 1/x - (d) d:
**Z**→**Z**where d(x) = x ⋅ |x| - (e) e:
**N**×**N**→**Z**where e(x, y) = 6x - 3y

- (a) a:
**Question 2 (15):**Let r be the proposition "it is raining", let c be "I go to class today", let h be "I did my homework", let b be "I go bicycling today", and let g be "I will have a great day". Write PropCalc formulas to express each of the following. Then convert each of your formulas to CNF. (Recall that CNF is conjunctive normal form: and's of or's of literals.)- (a) If I've done my homework and I go to class then I will have a great day.
- (b) If I go bicycling and do not go to class then I will have a great day, but I won't go bicycling if it is raining.

**Question 3 (10):**Define {a, b}^{*}= {λ, a, b, aa, ab, ba, bb, aaa,...} to be the set of all**finite strings**from the alphabet {a, b}. A finite string is a finite sequence of symbols, including λ which is the**empty string**. Argue that {a, b}^{*}is countably infinite, i.e., it has the same cardinality as**N**.**Question 4 (15):**Use Euclid's algorithm to solve the following problems:- (a) Compute gcd(119, 15).
- (b) Express gcd(119, 5) as a linear combination of 119 and 15.
- (c) If 15 has a multiplicative inverse modulo 119, compute it. If not, explain why not.

**Question 5 (15):**Use Euclid's algorithm to solve the following problems:- (a) Compute gcd(253, 66).
- (b) Express gcd(253, 66) as a linear combination of 253 and 66.
- (c) If 66 has a multiplicative inverse modulo 253, compute it. If not, explain why not.

**Question 6 (10):**Compute 9^{9}mod 11. Please use the repeated-squaring method taught in class.**Question 7 (10):**Find the smallest natural number that satisfies the following congruences: x ≡ 3 (mod 4), x ≡ 4 (mod 5), and x ≡ 1 (mod 7). Please use the method taught in class.

Last modified 26 February 2013