Exam Practice Problems

Transcription

1 Math 231 Exam Practice Problems WARNING: This is not a sample test. Problems on the exams may or may not be similar to these problems. These problems are just intended to focus your study of the topics. Note: The final is cumulative. MIDTERM I: (1) Which of the following are propositions? If it is a proposition, determine its truth value: (a) Is Jack 20 years old? (b) Do your homework! (c) This statement is not a proposition. (d) x < 2. (e) If 3 < 2 then 1 R. (f) For all finite sets X and Y, it follows that X Y = X Y. (g) Does induction require the base case(s)? (h) x R, y R, y 4 = x. (i) x R, y R, y 4 > x. (j) y Q, x R, y < x (k) y Q, x R, y < x 2. (l) y Q, x R, y < x 3.

2 (2) Let U = {n Z 0 n 9}. Let A = {2, 3, 5, 7} and B = {2, 4, 6, 8}. (a) Find A B. (b) Find A B. (c) How many elements does the power set P(A) have? (3) Assume p and q are true propositions, while r is a false propositions. Find the truth value of the following propositions. (a) (p q) (p (q r)). (b) ((p q) r) (p q).

3 (4) Assume P (x) and Q(x) are propositional functions on the domain D and r D. Suppose it is given that P (r) is true and Q(r) is false. For each quantified proposition, first determine if we have enough information to give it a true value. If there is enough information, report the truth value. (a) x, P (x). (b) x, Q(x). (c) x, y, Q(y) P (x). (d) x, y, P (x) Q(y).

4 (5) Let the domain of discourse be the real numbers. Determine the truth value of the following quantified propositions. (a) x, (x < 2 x 2 < 4). (b) x, (x < 2 x 2 < 4). (6) Let the domain of discourse be the direct product of the real numbers with the real numbers. Determine the truth value of the following quantified propositions. (a) x, y, x 2 + y < 0. (b) x, y, x 2 + y < 0. (7) Negate the quantified proposition in Problem (6), Part (a).

5 (8) Prove the following propositions or find a counterexample. (a) If m is an even integer and n is an odd integer then mn is even. (b) Let m, n be integers. If mn is an even integer then m and n are even integers. (9) Use induction to prove the following proposition: For all integers n 4, n 2 2 n.

6 (10) Use induction to prove the following proposition: For all integers, n 1, (2n 1) 2 = 4n3 n. 3 (11) Use induction to prove the following proposition: n ( ) 2 n(n + 1) n N, i 3 =. 2 i=1 (12) Using induction prove that n N, 5 n 1 is divisible by 4.

7 (13) Prove the following proposition or find a counterexample: For all sets A, B and C, if A C = B C then A = B. (14) Prove the following proposition or find a counterexample: For all sets A, B and C, if C and A C = B C then A = B.

8 MIDTERM II: (15) True or False? (a) If f g is one-to-one, then g is one-to-one. (b) If f g is onto, then g is onto. (c) The function f : N N given by f(n) = n 2 is injective (1-1). (d) The function f : N N given by f(n) = n 2 is surjective (onto). (e) The function f : N N given by f(n) = n 2 is bijective. (f) There are 60 rearrangements of the word AP P LE.

9 (16) How many 5-card deals from a standard deck of 52 contain at least one card from each of the four suits? (17) In how many ways can 3 boys and 5 girls stand in a line such that no two boys are standing next to each other?

10 (18) Suppose 8 fish are to distributed in 3 distinct tanks. In how many ways can this be done if... (a)...the fish are distinguishable? (b)...the fish are indistinguishable? (c)...the fish are indistinguishable and each tank must get a fish? (19) How many non-negative integer solutions are there to x 1 + x 2 + x 3 = 20? (20) Show that a nonempty set has the same number of subsets with an odd number of elements as it has with an even number of elements.

11 (21) Assume f : Z A given by f(x) = 2x 5 is onto. Determine the set A. (22) Determine if g : R (0, 1] given by f(x) = 1 1+x 2 is an injection (1-1), a surjection (onto), and a bijection. (23) Is the binary relation {(1, 1), (2, 3), (3, 2), (3, 3), (4, 4), (5, 6), (6, 6)} on N symmetric? (24) Is the binary relation {(2, 2), (2, 3), (3, 3), (3, 4), (4, 4), (5, 6), (5, 7), (6, 7)} on N anti-symmetric?

12 (25) Let f : N N be given by f(1) = 1 and for n 2, f(n) = n 1. Prove that f is a surjection (onto) but not a bijection. (26) Let f : N N be given by f(n) = n + 1. Prove that f is an injection (1-1) but not a bijection. (27) Consider these definitions: A function f : R R is strictly increasing if x, y R, x > y f(x) > f(y) and strictly decreasing if x, y R, x > y f(x) < f(y). (a) Show that if f : R R is strictly increasing and x R, f(x) > 0 then g(x) = 1 f(x) decreasing. is strictly (b) Show that if f : R R is strictly increasing then g(x) = f(1 x) is strictly decreasing.

13 (28) Consider the binary relation R on R given by (x, y) R if x 2 = y 2. Determine if this relation is reflexive, symmetric, transitive, and anti-symmetric. (29) Consider the binary relation R on N given by (m, n) R if m divides n. Determine if this relation is reflexive, symmetric, transitive, and anti-symmetric. (30) Consider these definitions: A function f : R R is even if x R, f( x) = f(x) and odd if x R, f( x) = f(x). Suppose f : R R and g : R R. Show that h(x) = (f g)(x) is odd. Show that k(x) = f(x)g(x) is even. (31) Let A, B be nonempty sets. Let f : A B and S B. The inverse image of S under f is f 1 (S) = {x A f(x) S}. Prove that f 1 (S) = f 1 (S).

14 FINAL: (32) True or False? (a) A connected graph on n vertices must have at least n 1 edges. (b) The sum of all the vertex degrees in a graph is even. (c) Every Euler cycle is a Hamiltonian cycle. (d) There exists a graph whose vertices have the following degrees: 2, 2, 1, 1, 1, 1. (33) Your friend says that his favorite (simple) graph has the following degrees: 1, 1, 1, 2, 3, 4, 4, 5. Does such a graph exist? If so, draw it. If not, justify. (34) Draw a graph such that... (a)...there is a Hamiltonian cycle but no Euler cycle. (b) There is an Euler cycle but no Hamiltonian cycle.

15 (35) Draw a simple undirected graph with the following degrees or explain why such a graph does not exist: 1, 1, 2, 3, 3, 6. (36) Draw a graph with the following adjacency matrix:

16 (37) Let G = (V, E) be a simple non-empty graph. Show that the relation R on V such that (u, v) R if and only if there is an path between u and v is symmetric, and transitive. (38) Show that in a simple graph with at least two vertices there must be (at least) two vertices of the same degree.

17 (39) Apply Dijkstra s algorithm to find the length of the shortest path between vertices u, v in the graph. Is the path unique?

18 (40) Apply Kruskal s algorithm to find a minimal spanning tree in the graph.