Download Relations - Mathematics - Exam and more Exams Mathematics in PDF only on Docsity!
MATHEMATICAL TRIPOS Part IA
Monday, 1 June, 2009 1:30 pm to 4:30 pm
PAPER 4
Before you begin read these instructions carefully.
The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt all four questions from Section I and at most five questions from Section II. In Section II, no more than three questions on each course may be attempted.
Complete answers are preferred to fragments.
Write on one side of the paper only and begin each answer on a separate sheet.
Write legibly; otherwise you place yourself at a grave disadvantage.
At the end of the examination:
Tie up your answers in separate bundles, marked A and E according to the code letter affixed to each question. Include in the same bundle all questions from Section I and II with the same code letter.
Attach a completed gold cover sheet to each bundle.
You must also complete a green master cover sheet listing all the questions you have attempted.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Gold cover sheets None Green master cover sheet
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
SECTION I
1E Numbers and Sets Let R 1 and R 2 be relations on a set A. Let us say that R 2 extends R 1 if xR 1 y implies that xR 2 y. If R 2 extends R 1 , then let us call R 2 an extension of R 1.
Let Q be a relation on a set A. Let R be the extension of Q defined by taking xRy if and only if xQy or x = y. Let S be the extension of R defined by taking xSy if and only if xRy or yRx. Finally, let T be the extension of S defined by taking xT y if and only if there is a positive integer n and a sequence (x 0 , x 1 ,... , xn) such that x 0 = x, xn = y, and xi− 1 Sxi for each i from 1 to n.
Prove that R is reflexive, S is reflexive and symmetric, and T is an equivalence relation.
Let E be any equivalence relation that extends Q. Prove that E extends T.
2E Numbers and Sets (a) Find integers x and y such that
9 x + 12y ≡ 4 (mod 47) and 6 x + 7y ≡ 14 (mod 47).
(b) Calculate 43^135 (mod 137).
3A Dynamics and Relativity A rocket moves vertically upwards in a uniform gravitational field and emits exhaust gas downwards with time-dependent speed U (t) relative to the rocket. Derive the rocket equation m(t) dv dt
= −m(t)g ,
where m(t) and v(t) are respectively the rocket’s mass and upward vertical speed at time t. Suppose now that m(t) = m 0 −αt, U (t) = U 0 m 0 /m(t) and v(0) = 0. What is the condition for the rocket to lift off at t = 0? Assuming that this condition is satisfied, find v(t). State the dimensions of all the quantities involved in your expression for v(t), and verify that the expression is dimensionally consistent. [ You may assume that all speeds are small compared with the speed of light and ne- glect any relativistic effects. ]
Part IA, Paper 4
SECTION II
5E Numbers and Sets (a) Let A and B be non-empty sets and let f : A → B. Prove that f is an injection if and only if f has a left inverse. Prove that f is a surjection if and only if f has a right inverse. (b) Let A, B and C be sets and let f : B → A and g : B → C be functions. Suppose that f is a surjection. Prove that there is a function h : A → C such that for every a ∈ A there exists b ∈ B with f (b) = a and g(b) = h(a). Prove that h is unique if and only if g(b) = g(b′) whenever f (b) = f (b′).
6E Numbers and Sets (a) State and prove the inclusion–exclusion formula. (b) Let k and m be positive integers, let n = km, let A 1 ,... , Ak be disjoint sets of size m, and let A = A 1 ∪... ∪ Ak. Let B be the collection of all subsets B ⊂ A with the following two properties: (i) |B| = k; (ii) there is at least one i such that |B ∩ Ai| = 3. Prove that the number of sets in B is given by the formula
⌊ ∑k/ 3 ⌋
r=
(−1)r−^1
k r
m 3
)r( n − rm k − 3 r
7E Numbers and Sets Let p be a prime number and let Zp denote the set of integers modulo p. Let k be an integer with 0 6 k 6 p and let A be a subset of Zp of size k.
Let t be a non-zero element of Zp. Show that if a + t ∈ A whenever a ∈ A then k = 0 or k = p. Deduce that if 1 6 k 6 p − 1, then the sets A, A + 1,... , A + p − 1 are all distinct, where A + t denotes the set {a + t : a ∈ A}. Deduce from this that
(p k
is a multiple of p whenever 1 6 k 6 p − 1.
Now prove that (a + 1)p^ = ap^ + 1 for any a ∈ Zp, and use this to prove Fermat’s little theorem. Prove further that if Q(x) = anxn^ + an− 1 xn−^1 +... + a 1 x + a 0 is a polynomial in x with coefficients in Zp, then the polynomial (Q(x))p^ is equal to anxpn^ + an− 1 xp(n−1)^ +... + a 1 xp^ + a 0.
Part IA, Paper 4
8E Numbers and Sets Prove that the set of all infinite sequences (ǫ 1 , ǫ 2 ,.. .) with every ǫi equal to 0 or 1 is uncountable. Deduce that the closed interval [0, 1] is uncountable. For an ordered set X let Σ(X) denote the set of increasing (but not necessarily strictly increasing) sequences in X that are bounded above. For each of Σ(Z), Σ(Q) and Σ(R), determine (with proof) whether it is uncountable.
9A Dynamics and Relativity Davros departs on a rocket voyage from the planet Skaro, travelling at speed u (where 0 < u < c) in the positive x direction in Skaro’s rest frame. After travelling a distance L in Skaro’s rest frame, he jumps onto another rocket travelling at speed v′^ (where 0 < v′^ < c) in the positive x direction in the first rocket’s rest frame. After travelling a further distance L in Skaro’s rest frame, he jumps onto a third rocket, travelling at speed w′′^ (where 0 < w′′^ < c) in the negative x direction in the second rocket’s rest frame. Let v and w be Davros’ speed on the second and third rockets, respectively, in Skaro’s rest frame. Show that
v = (u + v′)
uv′ c^2
Express w in terms of u, v′, w′′^ and c. How large must w′′^ be, expressed in terms of u, v′^ and c, to ensure that Davros eventually returns to Skaro? Supposing that w′′^ satisfies this condition, draw a spacetime diagram illustrating Davros’ journey. Label clearly each point where he boards a rocket and the point of his return to Skaro, and give the coordinates of each point in Skaro’s rest frame, expressed in terms of u, v, w, c and L. Hence, or otherwise, calculate how much older Davros will be on his return, and how much time will have elapsed on Skaro during his voyage, giving your answers in terms of u, v, w, c and L. [ You may neglect any effects due to gravity and any corrections arising from Davros’ brief accelerations when getting onto or leaving rockets. ]
Part IA, Paper 4 [TURN OVER
12A Dynamics and Relativity (a) A particle of charge q moves with velocity v in a constant magnetic field B. Give an expression for the Lorentz force F experienced by the particle. If no other forces act on the particle, show that its kinetic energy is independent of time.
(b) Four point particles, each of positive charge Q, are fixed at the four corners of a square with sides of length 2a. Another point particle, of positive charge q, is constrained to move in the plane of the square but is otherwise free. By considering the form of the electrostatic potential near the centre of the square, show that the state in which the particle of charge q is stationary at the centre of the square is a stable equilibrium. Obtain the frequency of small oscillations about this equilibrium.
[The Coulomb potential for two point particles of charges Q and q separated by distance r is Qq/ 4 πǫ 0 r.]
END OF PAPER
Part IA, Paper 4
