University of Warwick

Coventry

First Year Maths Past Papers And Shizzle Brap Brap

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Reply 80

TheTallOne

Burn.

Foundations 2008, Q3: What is the usual name for the set of all equivalence classes?

Spoiler

That stuff isn't in the lecture notes :no:

EDIT: FML it is :emo:

. It's in the Philosophical Remarks section which 'Does not introduce anything rigorous or examinable'.

University of Warwick

Coventry

Reply 81

Kolya

Hathlan

I wrote up Q1 foundations 2006 then these people who were in my house wanted to look at pictures of horse boxes and I had one line left to do and they closed the window. So everyone can go die.

I hate to be the one to say this, but Ctrl+Shift+T brings closed tabs back up in Chrome and Opera...

Reply 82

Hathlan

Foundations 2006

Question 1
(a) State the Fundamental Theorem of Arithmetic

(b)

7\mathbb{Z} \cap 3\mathbb{Z}

is an additive subgroup of

\mathbb{Z}

. Which additive subgroup is it?

(c) State (without proof) De Morgan's Laws for sets A, B and their complements.

(d) Which of the prepositions (i)-(iv) is logically equivalent to

P \rightarrow Q

(i)

\lnot (P \lor Q)

(ii)

P \lor (\lnot Q)

(iii)

(\lnot P) \lor Q

(iv)

(\lnot P) \lor (\lnot Q)

(e) Which of the factors (x+2), (x-3) and (x-1) divide the polynomial

x^4-5x^3-13x^2+77x-60

without remainder?

(f) Prove, for integers

1 \leq k \leq n

, that

\begin{pmatrix} n \\ k-1 \end{pmatrix} + \begin{pmatrix} n \\ k \end{pmatrix} = \begin{pmatrix} n+1 \\ k \end{pmatrix}

[You may assume that

\begin{pmatrix} n \\ k \end{pmatrix} = \frac{n!}{k!(n-k)!}

(g) Give examples of well-defined functions

f: \mathbb{Z} \rightarrow \mathbb{Z}

which are:
(i) surjective but not injective;
(ii) injective but not surjective;
(iii) neither injective nor surjective;
(iv) bijective.

(h) What precisely does it mean to say that the cardinality of two finite sets, A and B, is equal?

(i) If

\alpha

is the permutation (1234) and

\beta

is the permutation (125) then write the compostion

\alpha\beta

in disjoint cycle notation.

(j) What conditions must a set G with a binary operation satisfy in order to be a group?

(k) Calculate

17^{18}mod19

Spoiler

Reply 83

Hathlan

Foundations 2006

Question 2
(a) Prove by induction:

1^2 + 2^2+5^2+... (2n+1)^2 = \frac{1}{3}(n+1)(2n+1)(2n+3)

Spoiler

(b) Find the highest common factor d=hcf{720, 5670}. Find integers a and b such that d= 720a+5670b. Show your working clearly (for both parts of this question).

Spoiler

Reply 84

Totally Tom

Hathlan

brapbrap

all seems good.
http://detexify.kirelabs.org/classify.html try this for your symbol needs.

\displaystyle\mapsto

Reply 85

Hathlan

Foundations 2006

Question 3
(a) What is the remainder on division of

(x^3+3x^2+2x+1)

(x^2+2x-3)

Spoiler

(b)

(x^3+3x^2+2x+1)

is exactly divisble by

(x^2+bx-c)

if and only if two equations in b and c are satisfied. Find the equations. [You do not need to find the values of b and c that satisfy them.]

Spoiler

Reply 86

Totally Tom

Hathlan

seems kk.

thread name change :biggrin:

And check out your new photo :smile:

Reply 87

TheTallOne

Kolya

I hate to be the one to say this, but Ctrl+Shift+T brings closed tabs back up in Chrome and Opera...

And Firefox and IE 8. :h:

2008 is a

paper with a question on both groups and equivalence relations.

Reply 88

Totally Tom

TheTallOne

And Firefox and IE 8. :h:

2008 is a

paper with a question on both groups and equivalence relations.

Geddit done. You said you had that paper covered.

Reply 89

Hathlan

Foundations 2006

Question 4

(a) Define the compostion g o f of two functions

f:A \rightarrow B

and

g:B \rightarrow C

Spoiler

(b) Prove that if f and g are injective, then g o f is also injective.

Spoiler

(c) Prove that if g o f is injective then f is injective, but give an example (of functions f, g and sets A, B, C) to show that g o f can be injective without g being injective.

Spoiler

(d) Outline the proof (without assuming anything about the cardinality of the two sets involved) that there is no surjection f from the natural numbers to the unit interval

(0,1) \subset R

Spoiler

Reply 90

Hathlan

2006 is a joke tbh. What can I say, I get luckeh. :merryxmas:

Reply 91

TheTallOne

I can't type latex on a phone

Reply 92

Hathlan

Analysis 2006

Question 1

(a) Give an example of an unbounded sequence that has a convergent subsequence.

(b) Give an example of an unbounded sequence that has no convergent subsequence.

(c) Give an example of a null sequence

(a_n)

such that the series

\sum a_n

does not converge.

(d) Find a value N such that

\begin{vmatrix} \frac{n^2}{n^2+1}-1 \end{vmatrix} \leq 10^{-3}

for every value of

n \geq N

.

(e) Let

(a_n)

be a sequence such that

|a_n| \leq 2^{-n}

for every

n>0

. Find a value N such that

\begin{vmatrix} \sum_{n=N}^ \infty a_n \end{vmatrix} \leq 10^{-3}

(f) Give an example of a sequence which is not a Cauchy sequence but has the property that for every

\epsilon > 0

and every

N>0

there exists

n>N

and

m>2m

such that

\begin{vmatrix} a_n-a_m \end{vmatrix} \leq \epsilon

.

(g) Give an example of two sequences

(a_n)

and

(b_n)

such that the sequence

(c_n)

defined by

c_n=a_n+b_n

converges to 1 but neither

(a_n)

(b_n)

converge.

(h) Give an example of a sequence

(a_n)

that tends to infinity but is neither increasing nor eventually increasing.

(i) Give an example of a bounded sequence

(a_n)

such that

\frac{a_{n+1}}{a_n}

tends to 1 but

(a_n)

doesn't converge.

(j) Say whether the following statement is true or false. "For every real number a there exists

(a_n)

of irrational numbers such that

\lim_{n \rightarrow \infty} a_n=a

."

(k) Say whether the following statement is true or false. If it is false, give a counterexample. "Let

(a_n)

be a postive decreasing null sequence. The series

\sum_{n=1}^ \infty (-1)^na_n

can always be rearranged so that it converges to 1."

(l) Say whether the following statement is true or false. If it is false, give a counterexample. "If a sequence

(a_n)

does not converge to 0, then there exists

\epsilon>0

such that

|a_n|> \epsilon

for every n."

(m) Say whether the following statement is true or false. If it is false, give a counterexample. "Let f be a postive decreasing function such that