STUDY GUIDE FOR THE WRECKONING. 1. Combinatorics. (1) How many (positive integer) divisors does 2940 have? What about 3150?

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1 STUDY GUIDE FOR THE WRECKONING. Combinatorics Go over combinatorics examples in the text. Review all the combinatorics problems from homework. Do at least a couple of extra problems given below. () How many (positive integer) divisors does 2940 have? What about 350? (2) How many permutations of the English alphabet are there in which letters A and B (a) are (b) are not next to each other? (3) In how many ways can you arrange the letters of the word MATHEMATICS so that (a) the sequence AA does not appear in the arrangement? (b) the consonants remain in the original order? (4) (a) In how many ways can you make a 3-people committee out of a group of 6 people? (b) In how many ways can you make 5 different 3-people committees out of a group of 6 people? (5) (a) In how many ways can you give away 2 apples and 0 pears to 4 people so that each person gets at least one apple and one pear? (b) In how many ways can you give away 8 apples, 9 pears and 0 peaches to 3 people so that each person gets at least one piece of each fruit? (6) Iva is in a gardening store looking at pepper plants. There are 20 different varieties. She will get back home with 8 pepper plants. (a) In how many ways can she choose 8 plants of mutually distinct varieties? (b) In how many ways can she make her acquisition if she is not committed to getting 8 plants of mutually distinct varieties?

2 2 STUDY GUIDE FOR THE WRECKONING 2. Proof-writing Go over the proofs that 2 is irrational, and that there are infinitely many primes. Go through all the proof-writing feedback you got on your homework and exam(s). () Show, without using the Fundamental Theorem of Arithmetic, that if a and c are two coprime integers (that is, GCD(a, c) = ) and if the product bc is divisible by a (that is, a bc), then b must be divisible by a. (2) Show, without using the Fundamental Theorem of Arithmetic, that if a product of two integers is divisible by a prime p then at least one of the integers is divisible by p. (3) Prove that p, for p a prime integer, is irrational. (4) Let n > 2 be an integer. Prove that there is a prime number p strictly between n and n!: n < p < n!. (5) Let p, p 2,..., p n be n distinct prime numbers. Show that is never an integer. p + p p n (6) Let p be a prime, and let 0 < k < p be an integer. Prove that the binomial coefficient ( p k) is divisible by p. (7) Show that for all integers a and b, and all prime numbers p the number (a + b) p a p b p is divisible by p. (8) Integers a, b and c are the sides of a right-angled triangle. Prove that at least one of a, b and c is divisible by 3. (9) Let a, b and c be three integers such that a 2 3b 2 = c 2. (a) Show that 3 (c a) or 3 (c + a); (b) Show that at least one of a and b has to be even. (0) Use the Fundamental Theorem of Arithmetic to show that if the total number of divisors of a positive integer n is odd, then the number n must be a perfect square. () (a) Show that ( ) n + k ( ) n = k ( ) n + k for all k n. (b) Use mathematical induction to prove the Binomial Theorem n ( ) n (x + y) n = x n k y k. k k=0 (2) Use the principle of mathematical induction to prove that the following identity holds for all integers n. (2n + ) + (2n + 3) + (2n + 5) (4n ) = 3n 2.

3 STUDY GUIDE FOR THE WRECKONING 3 (3) Use mathematical induction to show that for integers k, k 2,..., k n the product leaves remainder after division by 4. (4k + )(4k 2 + )...(4k n + ) (4) In calculus we discuss the Product Rule i.e. a theorem regarding the derivative of the product of two functions. The theorem states if f and g are two differentiable functions, so is their product f g. Furthermore, the derivative of f g is equal to: (f g) = f g + f g. Often times, however, we need a result that can handle a product of many functions. Use mathematical induction and the Product Rule as stated above to prove the Generalized Product Rule: (f f 2... f n ) = f f 2... f n + f f 2... f n f f 2... f n for differentiable functions f, f 2,..., f n. (5) Prove that 2 n 2n + for all integers n 3. (6) Let x be some real number with 0 < x <. Show that ( + x) n nx n for all integers n. (7) Use mathematical induction to prove that for all natural numbers n 2 we have (n ) 3 < n4 4. (8) Show that for all n the number 7 n 6n is divisible by Number Theory Review the Euclidean Algorithm for finding the greatest common divisor of two numbers. Make sure you know the exact statements of the Division Algorithm, the GCD theorem from Section 2.4 of your textbook and the Fundamental Theorem of Arithmetic. Review the process for finding some solutions of the Diophantine equations mx + ny = d. () (a) Find the prime number decomposition of 2730 and 350. (b) Find GCD(2730, 350) using the Euclidean Algorithm. (c) Express the above GCD as an integer linear combination of 2730 and 350. (d) Find the GCD and the LCM of 2730 and 350 using the Fundamental Theorem of Arithmetic. (2) Find at least one pair (x, y) of integer solutions of the equation 53x + 42y =.

4 4 STUDY GUIDE FOR THE WRECKONING 4. Logic () Express the following using only the logical operations, and, and simplify as far as possible. The variables p, q, r are some statements. (a) (p q) (b) (p q) (c) (p q) (p r) (d) q (q p) (e) (p q) (q p) (2) Please negate the following statements. (a) (m, n) N 2, m 2 = 2n 2 (b) n N, n 3 n {, 3} (c) n N, (d) x > 0, y > 0, 3 n 2 3 n y < x (e) C > 0, n N, e n < C n! (f) ε > 0, N N, n N, (g) ε > 0, δ > 0, x R, (3) Please prove or disprove the following: (a) x R, x x+ < ; (b) C N, n N, n + Cn; (c) ε > 0, n N, n < ε; (d) n N, ε > 0, n < ε; (e) ε > 0, n N, x n n 2 < ε. n > N n < ɛ x < δ x 2 < ɛ 5. Set Theory () Prove (or disprove) each of the following statements of set theory. The set I is some index set, possibly infinite and possibly even uncountable. (a) (A B) C = (A C) (B C) for all sets A, B, C; (b) ( λ I A ) λ B = λ I (A λ B) for all sets B and A λ with λ I; (c) (A B) (C D) = (A C) (B D) for all sets A, B, C, D; (d) A (B C) = (A B) (A C) for all sets A, B, C; (e) A ( λ I A ) λ = λ I (A A λ) for all sets A and A λ with λ I; (f) (A B) C = A (B C) for all A, B, C; (g) A (A B) = B; (h) P(A) P(B) = P(A B) for all A, B;

5 STUDY GUIDE FOR THE WRECKONING 5 (i) P(A) P(B) = P(A B) for all A, B; (j) (A B) C = (A C) (B C) for all A, B, C. (2) Find, with proof, the following unions. Unless specifically stated otherwise, the problems use the interval notation encountered in your pre-calculus classes: (a) n N ( n 2 +, + ) (b) ( n N, n (c) λ R {λ2 } ) (d) ϱ R {(x, y) R2 x 2 + y 2 = ϱ 2 } (a, b) = {x R a < x < b}, (3) Let U S and V T be such that S U = T V. Use the method of element chasing to show: (a) S V = T U; (b) S V = T U. (4) Let A B and D C. Use the method of element chasing to prove that (A C) (B D) B C. etc. 6. (Equivalence) Relations () Consider the relation D on R (i.e. consider D R R) such that x D y xy Q. (a) Is the relation reflexive? Prove your claim. (b) Is the relation symmetric? Prove your claim. (c) Is the relation antisymmetric? Prove your claim. (d) Is this an equivalence relation? (2) Let be an equivalence relation on a set A, and let a, b A be such that a b. Use the method of element chasing to show that a = b. (3) Let be an equivalence relation on a set A, and let a, b A. Show that if (a, b) then a b =. (4) Prove that the relation on Z given by: n m 5 (n m) is an equivalence relation. Then describe the equivalence classes 0, and 2. (5) Let A be the set of all positive real numbers, i.e. let A = (0, + ). Prove that the relation on A given by x y y x Q is an equivalence relation.

6 6 STUDY GUIDE FOR THE WRECKONING (6) Prove that the relation on R 2 {(0, 0)} given by (x, y) (z, w) ( λ R {0}, (z, w) = (λx, λy)) is an equivalence relation. Then describe the equivalence classes (,), (0,) and (,0). (7) Let G be a graph, and let V (G) be the set of its vertices. Define the relation on V (G) by the following: for two vertices v and w write v w if v = w or if there exists a walk starting at v and ending at w. Prove that is an equivalence relation. How do you visualize the equivalence classes of this equivalence relation? () Let f : N 2 N 2 be given by: Is f -? Is f onto? Prove your claims. (2) Consider the function f : R 2 R given by 7. Functions f ( m, n ) = (GCD(m, n), LCM(m, n)). f(x, y) := x + y. (a) Is this function onto? If not, find its range. (b) Is this function -? If not, restrict its domain so that it becomes an injection. (3) Let P be the set of all polynomials in one variable, and let D : P P be the derivative operator, D(P ) = P. (So for example, D(x 3 2x 2 + x) = 3x 2 4x +.) (a) Prove or disprove that D is injective; (b) Prove or disprove that D is surjective. (4) (a) Show that the composition of two bijections is a bijection. (b) If the composition of two functions is a bijection, do the individual functions in the composition have to be bijections? (5) Verify that the function F : R 2 R 2 given by is a bijection. What is its inverse F? (6) Let H denote the upper half-plane F (x, y) := ( x y, x 2y) H := {(x, y) R 2 y > 0}. Prove that the function F : H H given by is a bijection. F (x, y) = ( x x 2 + y 2, ) y x 2 + y 2 (7) Find a bijection between the intervals (0, 2) and (, 5) on the real number line. Prove your claim!

7 STUDY GUIDE FOR THE WRECKONING 7 (8) Prove or disprove each of the following. Employ the method of element chasing whenever possible. (a) Let f : X Y and let A, B X; then f(a B) = f(a) f(b). (b) Let f : X Y and let A, B X; then f(a B) = f(a) f(b). (c) Let f : X Y and let C, D Y ; then f (C D) = f (C) f (D). (d) Let f : X Y and let C, D Y ; then f (C D) = f (C) f (D). (9) Two simple graphs G = (V, E) and G = (V, E ) are said to be isomorphic if there exist bijections F : V V and F 2 : E E such that for all {v, w} E we have {F (v), F (w)} E. In such a case we write G = G. Show that = is an equivalence relation on simple graphs. 8. Graph Theory Review terminology: simple graph, multigraph, subgraph, degree, degree sequence, walk, trail, path, circuit, cycle, connected, connected component, Eulerian circuit, Eulerian trail, Eulerian graph, semi-eulerian graph, Hamiltonian cycle, Hamiltonian graph, planar graph, bipartite graph, K n, K m,n, homeomorphic, tree, spanning tree, minimal spanning tree. Review results: The Handshake Theorem, the Eulerian Graph Theorem, Ore s Theorem, Euler s Theorem about planar graphs, Kuratowski s Theorem, and the following algorithms: () Algorithm for finding an Eulerian circuit; algorithm for finding an Eulerian trail. (2) Prim s and Kruskal s Algorithms for finding minimal spanning trees. (3) Sorting algorithm based on binary trees. () Show that in a simple graph there have to be two vertices of the same degree. (Hint: Pigeonhole Principle.) (2) Is there are simple graph whose degree sequence is a) 3, 3, 3, 2? b) 3, 3, 3, 3, 2? c) 3, 2,, 0? (3) The following questions have to do with the complete bipartite graph K n,n. (a) What is the number of vertices and edges in the complete bipartite graph K n,n? (b) What is the degree sequence for the complete bipartite graph K n,n? (c) What is the length of the shortest cycle in the complete bipartite graph K n,n? (d) For what value of n is the complete bipartite graph K n,n : (i) planar? Briefly explain your answer. (ii) Hamiltonian? Briefly explain your answer. (iii) Eulerian? Briefly explain your answer. (iv) semi-eulerian? Briefly explain your answer.

8 8 STUDY GUIDE FOR THE WRECKONING (4) The following questions have to do with the complete bipartite graph K 4,3. (a) What is the minimal number of edges one needs to add to K 4,3 in order to obtain an Eulerian graph? Please modify K 4,3 by adding this minimal amount of edges; then find an Eulerian circuit of the modified graph. (b) What is the minimal number of continuous strokes needed to draw K 4,3? Please indicate the strokes. (5) Suppose that a planar graph G has k connected components, v vertices and e edges. Into how many regions is a planar drawing of G dividing the plane? In[4]:= (6) Let G be a simple connected planar graph with v vertices and e edges. (a) Show that if v 3 then 3v e 6; (b) Show that if v 4 and AxesOrigin the shortest Ø 80, 0<D, cycle is of length at least 4, then 2v e 4. (7) Show that K 5 and K 3,3 are not planar. Show@ListLinePlot@Table@8Cos@2 * n * Pi ê 8D +.5, Sin@2 * n * Pi ê 8D +.5<, 8n, 0, 4<D, AxesOrigin Ø 80, 0<, PlotMarkers Ø AutomaticD, ListLinePlot@Table@8Cos@2 * H-3 nl * Pi ê 8D +.5, Sin@2 * H-3 nl * Pi ê 8D +.5<, 8n, 0, <D, AxesOrigin Ø 80, 0<D, ListLinePlot@ Table@8Cos@2 * H- + 3 nl * Pi ê 8D +.5, Sin@2 * H- + 3 nl * Pi ê 8D +.5<, 8n, 0, <D, AxesOrigin Ø 80, 0<D, ListLinePlot@Table@8Cos@2 * H2 + 3 nl * Pi ê 8D +.5, Sin@2 * H2 + 3 nl * Pi ê 8D +.5<, 8n, 0, <D, ListLinePlot@Table@8Cos@2 * H - 3 nl * Pi ê 8D +.5, Sin@2 * H - 3 nl * Pi ê 8D +.5<, 8n, 0, <D, AxesOrigin Ø 80, 0<D, ListLinePlot@Table@8Cos@2 * H + 3 nl * Pi ê 8D +.5, Sin@2 * H + 3 nl * Pi ê 8D +.5<, 8n, 0, <D, AxesOrigin Ø 80, 0<D, (8) Use Kuratowski s Theorem to show that the following graph is not planar. Please explain your solution in detail. 2.5 ListLinePlot@Table@8Cos@2 * H3 + 3 nl * Pi ê 8D +.5, Sin@2 * H3 + 3 nl * Pi ê 8D +.5<, 8n, 0, <D, AxesOrigin Ø 80, 0<DD Out[4]= (9) Use Kuratowski s Theorem to show 0.5 that.0 the following graph is 2.5not planar. Please explain your solution in detail. (0) Recall that we define trees as connected graphs without circuits. Now let G be a simple connected graph on n vertices. Show that the following are equivalent: (a) G is a tree; (b) G has n edges; (c) removing an edge of G disconnects G.

9 STUDY GUIDE FOR THE WRECKONING 9 () A disconnected graph with no circuits is called a forest. How many edges does a forest with k connected components and n vertices have? (2) Show that there are at least two vertices of degree in each tree. (3) Is it possible to have a connected graph with n vertices and n 2 edges? Prove your answer. (4) A fancy network of fast trains is to be built in Europe. The network is to include London, Paris, Rome, Berlin and Moscow. The distances in miles between the cities are described in the following table: L P R B M L P R B M a) Model the network on a weighted K 5 graph. (You are only expected to provide a complete drawing of the weighted graph.) b) The investors and the construction company decided to start by building the shortest network connecting the five cities. Please give advice on which cities to connect. c) What is the total length of your proposed network?