Fundamentals of Pure Mathematics - Problem Sheet

Fundamentals of Pure Mathematics - Problem Sheet ( ) = Straightforward but illustrates a basic idea (*) = Harder Note: R, Z denote the real numbers, integers, etc. assumed to be real numbers. In questions 1.1-1.4 all variables are 1 Mathematical Argument 1.1 ( ) Which of the following statements are true for real numbers x? (a) x = 1 implies x 2 3x + 2 = 0. (b) x = 1 if x 2 3x + 2 = 0. (c) x = 1 only if x 2 3x + 2 = 0. (d) If x = 1 then x 2 3x + 2 = 0. (e) x = 1 is a necessary condition for x 2 3x + 2 = 0. (f) x = 1 is a sufficient condition for x 2 3x + 2 = 0. (g) x > 2 implies x > 1. (h) x > 2 is a sufficient condition for x > 1. 1.2 (a) Give a careful deductive proof that if 2x 2 5x + 2 < 0 then x > 0 (b) Give a proof by contradiction of the same fact. (c) Decide whether the following statement is true or false, giving a proof or a counterexample. For all x > 0, 2x 2 5x + 2 > 0. 1.3 ( ) What does the following argument prove? Assume that x 2 5x + 6 = 0. Then (x 2)(x 3) = 0. so either x 2 = 0 or x 3 = 0. Thus either x = 2 or x = 3. 1.4 ( ) Mark the following statements true or false. (a) There exists x such that sin(x + π/2) = sinx. (b) For all x, sin(x + π/2) = sin x. (c) There exists x such that for all y, y x. (d) For all y, there exists x such that y x. (e) x such that x 2 = 2x. (f) x, x 2 2x. (g) x z y such that x + y = z. 1

1.5 Euler suggested the following formula for generating prime numbers: if n is a positive integer then n 2 + n + 41 is a prime. Verify this for some small values of n. Give a counter example to show that it is not in general true. [Hint: chose an n that makes n 2 + n a multiple of 41. - In fact it may be shown that no such polynomial formula can generate only primes.] 1.6 Show by induction, writing out your proof very carefully, that for each positive integer n 1 1.2 + 1 2.3 +... + 1 n(n + 1) = n n + 1. 2 Sets, Relations and Functions 2.1 Let X = {1,2,3} and Y = {3,4}. Find X Y, X Y, X Y, X \Y, Y \ X, (X Y ) X, P (X), {n + m : n X,m Y }. 2.2 For all n = 1,2,3,... let X n R be the open interval (1/n,2 + 1/n). Find 4[ X n, n=1 4\ X n, n=1 [ X n, n=1 \ X n. n=1 2.3 Prove that for all sets X,Y and Z (a) (X Y ) Z = (X Z) (Y Z), (b) (X \Y ) (X \ Z) = X \ (Y Z), (c) (X Y ) (Y X) = (X Y ) (X Y ). 2.4 Let X denote the number of elements in a finite set X. Prove that for all finite sets X,Y and Z X Y = X + Y X Y, X Y Z = X + Y + Z X Y Y Z Z X + X Y Z. 2.5 Which of the properties Reflexive, Symmetric and Transitive are satisfied by the following relations where x,y R? (a) x < y (b) x y (c) x y < 1 (d) x y is rational (e) x y is irrational (f) x y < 1 (g) x < y 2 (h) xy 0. 2

2.6 Show that the following are equivalence relations on Z and describe the equivalence classes. (a) 3 divides x y (b) 5 divides x + 4y (c) x 2 = y 2 2.7 Define a relation R on the Euclidean plane R 2 by (x 1,y 1 )R(x 2,y 2 ) if and only if x 1 + y 1 = x 2 +y 2. Show that R is an equivalence relation and describe the equivalence classes geometrically. 2.8 Let R and S be equivalence relations on a set X. Show that the relation T defined by xty if and only if xry and xsy is an equivalence relation. Give and example to show that xry if and only if xry or xsy is not an equivalence relation. 2.9 Let Z denote the non-zero integers. Define a relation R on Z Z by (a,b)r(c,d) if and only if ad = bc. Show that this is an equivalence relation and describe the equivalence classes. Does this remind you of anything? 2.10 Define a relation on the Euclidean plane R R as follows: (x,y) (a,b) means either x < a or both x = a and y b. Show that is an order relation, and describe (x, y) (a, b) geometrically. 2.11 ( ) Establish which of the following functions are injections, surjections and bijections. Find the inverses of any bijections. (a) f : R R f (x) = x 2 (b) f : R R f (x) = x 3 (c) f : Z Z f (x) = x 2 (b) f : Z Z f (x) = x 3 (e) f : Z Z f (x) = x + 7 (f) f : Z Z f (x) = x + 7 (g) f : R R f (x) = x 3 + 9x (h) f : R R f (x) = x 3 9x. 2.12 Let f : X Y be a function and A X Show that f (A B) = f (A) f (B), f (A B) f (A) f (B). Give and example to show that cannot be replaced by = in the second case. 3

2.13 Let R be the equivalence relation on Z defined by xry if 3 divides x y. Which of the following binary operations on Z induce a well-defined operation on the equivalence classes of R? In each case either give a short proof and write out the table giving the induced operation, or give an example to show the operation is not well-defined. (a) x y = x + 2y (b) x y = xy (c) x y = x + y. 2.14 Let R be the equivalence relation defined on Z Z by (a,b)r(c,d) if and only if ad = bc (see question 2.9). Show that the binary operation defined by (a,b) (p,q) = (aq + bp, bq) induces a well-defined operation on the equivalence classes. What does this remind you of? 2.15 ( ) Let f : X Y and g : Y Z. Show that if f and g are both injections then the composition g f is an injection and that if f and g are both surjections then the composition g f is an surjection. 2.16 [For those familiar with some group theory] Let (G,.) be a group and H a subgroup. Define a relation on G by taking xry to mean xy 1 H. Show that R is an equivalence relation. Show that the equivalence classes are the left cosets xh = {xy : y H}. Show that if H is an normal subgroup, i.e. if y 1 Hy = H for all y, then the induced operation on G/R is well-defined (the quotient group ). 4

3 Number systems 3.1 Show from the defining properties of a ring (R,+,.): (i) 0 is unique (i.e. if r is an element such that r + a = a for all a then r = 0) (ii) if x + a = 0 then x = a (iii) ( a) = a for all a R. 3.2 Check, using the definitions of + and in Q and arithmetic properties in Z that in Q, 0/1 is the zero, 1/1 is the identity, that a/b is the negative of a/b and b/a is the inverse of a/b provided a/b is non-zero. 3.3 Assuming the arithmetic properties of Z, show that in the rationals, (A1) a + b = b + a for a,b Q (A2) a + (b + c) = (a + b) + c for a,b,c Q (M2) a(bc) = (ab)c for a,b,c Q. 3.4 Assuming the corresponding properties of the integers, verify that (O2) a b,c d implies a + c b + d for a,b,c Q. 3.5 Show that if a Q (or indeed a belongs to any ordered field) then 0 a 2. 3.6 Prove that if a,b Q with 0 a b then a n b n for every positive integer n (where we define a n in the obvious way as an n-fold product) [Hint: induction]. Show that if a Q with 0 a there is at most one element r Q such that r n = a. 3.7 The aim of this question is to show how one could start just with the natural numbers N {1,2,3,...} and use equivalence classes to construct the integers Z: this is analogous to the way we constructed Q from Z. Here you may use and standard arithmetic properties of the natural numbers but not subtraction or properties of negative numbers. (a) Show that defines an equivalence relation on N N. (a,b)r(c,d) if and only if a + d = b + c In the usual way, we write [a,b] for the equivalence class under R that contains (a,b), and call the set of these equivalence classes Z. Define operations + and on the equivalence classes in terms of arithmetic operations on N by [a,b] + [c,d] = [a + c,b + d], [a,b] [c,d] = [ac + bd,ad + bc]. (b) Show that + is a well-defined operation on these equivalence classes (i.e. equivalent elements map to equivalent elements). 5

(c) Show that, for all a,b,c N, [a,b] + [c,c] = [a,b] and [a,b] + [b,a] = [c,c]; Thus we may think of [c,c] as the zero of Z and [b,a] as the negative of [a,b]. (d) We can now verify all the conditions for Z to be a ring. Show, for example, that the distributive law holds in Z, that is [a,b]([c,d] + [e, f ]) = [a,b][c,d] + [a,b][e, f ]. (e) To check that N may be regarded as a subset of Z, define f : N Z by f (x) = [1,x + 1]. Show that f is an injection, such that f (x + y) = f (x) + f (y) (where addition is in the respective sets). Similary we can show that f respects multiplication and order, so we may identify {[1,a + 1] : a N} with N. 3.8 Consider the following sets of rational numbers: X = {x Q : 1 < x < 5}, Y = {x Q : 3 x 7}, U = {1, 12, 14,..., 12 n,... }. For each of the following sets determine its maximum, minimum, least upper bound and greatest lower bound (whenever they exist), and whether or not it is dense: 1. X; 2. Y ; 3. X Y ; 4. X Y ; 5. Y \ X; 6. X \Y ; 7. X Z; 8. Y \ Z; 9. U; 10. U \ X. 3.9 On the set Q of rationals define a relation R as follows: xry x y Z. Prove that this is an equivalence relation. If we try to define addition and multiplication on the set of equivalence classes by [x] + [y] = [x + y] and [x][y] = [xy], one of the is well-defined and the other isn t. Which is which? Show that if x Q then there is a positive integer n such that the n-fold sum [x] + [x] + + [x] = [0]. 3.10 Prove that if A is a cut then so is A. Moreover, prove that A is positive if and only if A is negative. 3.11 Prove that for arbitrary cuts A,B we have A + B = B + A and AB = BA. 3.12 Let A,B,C be arbitrary cuts. (i) Prove that AB > O if and only if A and B are of the same sign (i.e. A,B > O or A,B < O). (ii) Prove that A 2 0 for every cut A. 6

(iii) Prove that if A < B and C > O then AC < BC. [Hint: Use without proof that B A > O, and the distributive law.] (iv) Prove that if A > B > O and n N then A n > B n. (v) Prove that for any positive y R and any n 1 there exists at most one positive x R such that x n = y. 3.13 For each of the following combinations of conditions (C1), (C2), (C3) from the definition of a cut try to find a set satisfying it: (a) (C1), (C2) but not (C3); (b) (C3), but not (C1) nor (C2); (c) none; (d) (C2), (C3) but not (C1). 3.14 Verify the distributive law on C from the definitions of the operations on C: (D) a(b + c) = ab + ac for a,b,c C. 3.15 Derive the reverse triangle inequality for complex numbers, that z w z w for z,w C. 3.16 Show that for a polynomial p(z) = a n z n + + a 1 z + a 0 with real coefficients a i, that zeros occur in conjugate pairs, i.e. if p(z) = 0 then p(z) = 0. 7

4 Number theory 4.1 Use Euclid s algorithm to find solutions in integers x,y of the following equations: (a) 34x + 10y = 2 (1 solution) (b) 34x + 10y = 2 (all solutions) (c) 34x + 10y = 3 (all solutions) (d) 34x + 10y = 4 (all solutions) 4.2 Show that if the integers n and m are of the form 6k + 1 (i.e leave a remainder 1 on division by 6) then nm is also of the form 6k + 1. Hence show that there are infinitely many prime numbers of the form 6k +5. [Hint: Suppose p 1, p 2,..., p n are the only such primes and consider m = p 1 p 2... p n 1.] 4.3 Show that there are arbitrary long gaps between consecutive primes. [Hint: Show that there are arbitrarily long sequences of consecutive composite (i.e. non-prime) numbers - consider sequences starting n! + 2.] 4.4 Use Chebychev s prime number theorem to show that for every integer n there is a prime between n and 40n. Show that there are at least cn/logn primes in this range, for some positive constant c. 4.5 Show that the series p 1 p is divergent (where the sum is over all primes). [Hint: split the sum up as n=1 40 n <p 40 n+1 and use the estimate of the previous question.] 4.6 Show that if a 2 and a n + 1 is prime, then a is even and n is a power of 2. (Primes of the form 2 2n + 1 are called Fermat primes.) 4.7 Show that if n > 1 and a n 1 is prime, then a = 2 and n is prime. (Primes of the form 2 n 1 are called Mersenne primes.) 4.8 Let the positive numbers a, b be rational and s,t be irrational. Determine whether each of the following numbers are (i) always rational (ii) always irrational (iii) may be rational or irrational: (a) a + b; (b) s +t; (c) a + s; (d) ab; (e) as; (f) a; (g) s; 4.9 Show from the definition that 10 is irrational. 4.10 Show that the following numbers are irrational: (a) 2 + 5; (b) 2 + 3 5. 4.11 Show that is irrational. 1 + 1 (1!) 2 + 1 (2!) 2 + 1 (3!) 2 + 8

5 Countability and Cardinality 5.1 Let X and X be disjoint sets and let Y and Y be disjoint sets. Show that if X Y and X Y then X X Y Y. Show that {0,1, 2 1, 1 3, 4 1,...} { 1 2, 1 3, 1 4,...}. Deduce that [0,1] (0,1) (where [0,1] and (0,1) are the closed and open unit intervals). 5.2 Determine, giving brief reasons (you may quote any standard results), whether each of the following sets is countable or uncountable. 1. The set of prime numbers. 2. {a + bi : a,b N}. 3. Q Q = {(q,r) : q,r are rational}. 4. {x R : 0 < x < 10 100 }. 5. {n 2 : n N}. 6. {x R : x > 0}. 7. { m n R + : m,n N}. 5.3 Define f : N N N by f (m,n) = 2 m 3 n. Show that f is an injection. Deduce that N N is countable. Now define f : {finite subsets of N} N by f (A) = 2 m 13 m 25 m 37 m 411 m 5... where m k = 1 if k A and m k = 0 if k / A. Show that f is a well-defined injection. Deduce that the set of all finite subsets of N is countable. 5.4 Show that the unit interval (0,1) and the interior of the unit square (0,1) (0,1) have the same cardinality. [Hint: How can the decimal expansions of two numbers x, y (0,1) be mapped (bijectively) into the decimal expansion of a single number k (0,1)?] 5.5 Use the Schroeder Bernstein Theorem to prove that (0, 1) (open interval) and [0, 1] (closed interval) are similar (compare Question 5.1). 5.6 Determine the cardinality of each of the following sets: 1. The set of lines in the Euclidean plane R 2. 2. The set of all sequences of integers. 3. The set of all sequences of real numbers. 5.7 What are the cardinalities of the following sets: (a) the set of all injective mappings N N; (b) the set of all equivalence relations on N? 5.8 Prove the following statements: (i) (ii) If A = B and B < C then A < C. If A < B and B < C then A < C. 9