M1F Foundations of Analysis
Problem Sheet 9
1. Fix f : S T and suppose there exists g: T S such that f g = idT . Is f
(a) injective ?
(b) surjective ?
(c) bijective ?
Give a proof or counterexample fo
M1F Foundations of Analysis
Problem Sheet 6
1. In Mestelland all months have exactly 30 days. Much neater. Show that some
months start on a Monday.
2. Professor Mestel likes to have fun every 5 days a
M1F Foundations of Analysis
Problem Sheet 10
1. Fix S R with an upper bound, and suppose that S = and S = R. Give
proofs or counterexamples to the following statements.
(a) If S Q then sup S Q.
(b) If
M1F Foundations of Analysis
Problem Sheet 1
1. Let A be the set cfw_1, 3, 6, cfw_1, 6, Doncaster, cfw_1, X. Which of the following
statements are true and which are false ? (Just write T or F in each
M1F Foundations of Analysis
Problem Sheet 2
1. What is the biggest element of the set cfw_x R: x < 1 ? Justify your answer
carefully.
It does not exist. Suppose it did, call it m < 1. Let n = (m + 1)/
M1F Foundations of Analysis
Problem Sheet 4
1. Irrational Mestel tries to show his tutees that 12 3 is rational, by the
following argument.
12 3 = p/q, p, q N,
= 12 2 12 3 + 3 = p2 /q 2 ,
= 15 2 36
M1F Foundations of Analysis
Problem Sheet 3
1. Show by induction that 7n 3n is always divisible by 4.
Can you see the one-line proof that this true, not using induction ?
Clearly true for n = 0. Assu
M1F Foundations of Analysis
Problem Sheet 5
1. Professor Mestel is furious. That quadratic equation was on his desk when he
went to lunch, and now he cant find it anywhere. He can remember its roots
M1F Foundations of Analysis
Problem Sheet 6
1. In Mestelland all months have exactly 30 days. Much neater. Show that some
months start on a Monday.
30 and 7 are coprime, so there exist integers p, q s
M1F Foundations of Analysis
Problem Sheet 7
1. Let p be a prime, and fix a which is not equal to 0 or 1 modulo p. Prove that
1 + a + . . . + ap2 0 mod p.
By Fermats little theorem, ap1 1 mod p.
Theref
M1F Foundations of Analysis
Problem Sheet 8
1. For each relation below, state which of symmetric, reflexive, transitive and an
equivalence relation it is. ( x denotes the integer part of x, i.e. the l
M1F Foundations of Analysis
Problem Sheet 9
1. Fix f : S T and suppose there exists g: T S such that f g = idT . Is f
(a) injective ?
(b) surjective ?
(c) bijective ?
Give a proof or counterexample fo
M1F Foundations of Analysis
Problem Sheet 10
1. Fix S R with an upper bound, and suppose that S = and S = R. Give
proofs or counterexamples to the following statements.
(a) If S Q then sup S Q.
(b) If
KMB, 3/10/17
M1F Foundations of Analysis, Problem Sheet 1
1. Which of the following statements involving an integer x are true and which are false? Just
write T or F, and perhaps also one remark about
KMB, 3/10/17
M1F Foundations of Analysis, Problem Sheet 1 solutions.
1.
(a) F (x = 2 is also a root)
(b) T (it doesnt matter that x = 2 is a root here)
(c) F (x = 2 is a problem again)
(d) T (the two
KMB, 16/10/17
M1F Foundations of Analysis, Problem Sheet 2.
1. What are the following sets? Justify your answers.
S
(a) n=0 [n, n + 1).
S
(b) n=1 [1/n, 1].
S
(c) n=1 (n, n).
T
(d) n=1 (n, n).
2. Prove
M1F Foundations of Analysis
Problem Sheet 5
1. Professor Mestel is furious. That quadratic equation was on his desk when he
went to lunch, and now he cant find it anywhere. He can remember its roots
M1F Foundations of Analysis
Problem Sheet 7
1. Let p be a prime, and fix a which is not equal to 0 or 1 modulo p. Prove that
1 + a + . . . + ap2 0 mod p.
2. (Mini RSA.) Let p be a prime, and fix some
KMB, 24/10/14
M1F Foundations of Analysis, Problem Sheet 3.
1. Using only the 4 standard inequality facts (A1) to (A4), write down a proof that if 0 < x and
0 < y then 0 < x + y. Hint: youll only need