Practice Final. Read all the problems first before start working on any of them, so you can manage your time wisely

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1 PRINT your name here: Practice Final Print your name immediately on the cover page, as well as each page of the exam, in the space provided. Each time you are caught working on a page without your name printed on it, you will loose one point. This exam is closed book. You are only allowed to use one page of notes (double sided is fine) Your solution will be evaluated both for correctness and clarity. A poorly written solution won t get full credit even if correct. Read all the problems first before start working on any of them, so you can manage your time wisely Problem Points Score Total 74 1

2 1 Truth Table [5 points] Determine if the following statement is a tautology using the truth table method: (A (B C)) ((A B) (A C)) 2 Sets [5 points] Answer the following true/false questions. (Circle your answer). T F For all sets A, B, we have A B {x x A x B} T F For all sets A, B, we have A \ B = {x (x A) (x B)} T F For all sets A, B, we have A B = A + B T F For some sets A, B, we have A B = A + B T F For every sets A and B, A c B A B c 2

3 3 Relations [8 points] Consider the following relation R = {(a, b) a, b {0, 1, 2, 3, 4, 5} c Z.a 2b = 3c} Answer the following questions (circle the correct answer) and provide a brief explaination: Is R transitive? YES NO Is R symmetric? YES NO Is R reflexive? YES NO Is R an equivalence relation? YES NO 3

4 4 Logic (6 points) For each of the following statements Formulate the statement in English (without using mathematical symbols) using only words from the following list Write the logical negation of the statement (this time using only logical and mathematical notation) but without using the symbol. (For this part, you can use the fact that for any two integers x, y, the negation of x < y is x y.) Say whether the original statement is true. Statement: x Z. y Z.x < y Statement: y Z. x Z.x < y 4

5 5 Divisibility (8 points) In this problem, all variables range over the integers. Recall that x divides y (in symbols x y) if c.(y = c x). Formulate the following statement using the language of predicate logic (you can use the relation x y as an abbreviation), and prove that it is correct. Statement: For any x, y and z, if x divides both y and z, then for any a and b, x also divides ay + bz. Logic statement: Proof: 5

6 6 Logic (6 points) Formulate each of the following sentences using predicate logic and the basic predicates T(x)= x is a CSE20 student, S(x)= x is a surfer, F(x)= x is a frisbee player and L(x,y)= x likes y. 1. Alice, Bob and Charlie are CSE20 students. 2. All CSE20 students are either surfers or frisbee players. 3. No frisbee player likes to get up early in the morning. 4. All surfers like sunny weather. 5. Alice likes whatever Charlie dislikes, and dislikes whatever Charlie likes. 6. Charlies likes both getting up early in the morning and sunny weather. 6

7 7 Proof (6 points) Formulate the next statement in predicate logic using the same notation as in the previous problem. Then, prove that this statement using the assertions in the previous problem as premises. Some CSE20 student is a frisbee player, but not a surfer. Proof 7

8 8 Program correctness [8 points] Use an appropriate loop invariant to prove that the following program outputs 2 a 1 for all integer inputs a 0 (All variables in the program take integer values.) [ INPUT: a 0 ] x := 0 y := a while (y 0) x := 2 x + 1 y := y 1 end while output x Loop Invariant: Proof of Partial Correctness: 8

9 9 Recursion [8 points] Prove by induction that for all nonnegative integers a, b, the following recursive program computes the value (a + 2b) b!, where b! = b is the factorial function. MyProgram(a,b) = if (b == 0) then return a else return b MyProgram(a + 2, b 1) (Remember that in a proof by induction you need first to define a predicate P(n), and then prove that n 0.P (n).) Claim: Proof: 9

10 10 Monotone Functions [4 points] Let be a transitive relation over a set A. A function f: A A is called monotone if x, y A.(x y) (f(x) f(y)). Prove that if f: A A and g: A A are monotone functions, then f g: A A is also monotone. 11 Number systems [4 points] Answer each of the following question: The binary representation of (165) 10 is The decimal representation of (AF 1) 16 is (mod 5) equals The multiplicative inverse of 7 (mod 16) is 10

11 12 Counterexamples [6 points] Show that each of the following statements is false by giving an appropriate counterexample: If f: A B is injective and g: B C is surjective, then g f: A C is a bijection. All functions f: A A are surjective If f: A B is injective, then A and B have the same size 11