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 e coprime to (p 1).
(a) Show that there exists d such t
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
and but not the equation. He tries to cheer himself up
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 for each.
(You are strongly advised to try to draw a diag
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 and eat salad every 8 days. Show
that he sometimes has f
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 S R\Q then sup S R\Q.
(c) If S Z then sup S Z.
(d) The
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 case.)
(a) X A
(T)
(b) cfw_X A
(c) cfw_X A
(F)
(F)
(d)
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)/2. Then m = (m + m)/2 < (m + 1)/2 <
(1 + 1)/2 = 1 shows
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 = p2 /q 2 .
Since 36 = 6 is indeed rational, this looks
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. Assume true for n = k, then
7k+1 3k+1 = 7(7k 3k ) + 7.3k 3k
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
and but not the equation. He tries to cheer himself up
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 such that 30p + 7q = 1.
In fact by Euclid we can work th
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.
Therefore p divides ap1 1 = (a 1)(1 + a + . . . ap2 ).
But a
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 largest integer n x.)
(a) On R, x y x = y .
(b) On R, x
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 for each.
(You are strongly advised to try to draw a diag
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 largest integer n x.)
(a) On R, x y x = y .
(b) On R, x
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 = p2 /q 2 .
Since 36 = 6 is indeed rational, this looks
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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 case.)
(a) X A
(b) cfw_X A
(c) cfw_X A
(d) cfw_1, 6 A
(
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.
2. Let n be an integer. Prove carefully that if n2 is divisible by 3 then so is n.
(Hint: any integer can be written in t