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Maths 1 > Sem 1 Liebeck > Flashcards

Sem 1 Liebeck Flashcards

1
Q

equality

A

exactly same elements

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2
Q

P=>Q

A

P implies Q
if P then Q
Q if P
P only if Q

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3
Q

negation p bar

A

If P=>Q
then Q bar => P bar

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4
Q

negation of for all

A

there exists

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5
Q

for all x statements in the empty set

A

always true

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6
Q

there exists an x statements in the empty set

A

always false

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7
Q

number systems rules

A

a+b = b+a
a+(b+c)=(a+b)+c
a(b+c)=ab+ac

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8
Q

between any 2 rationals…

A

there is another rational

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9
Q

if a is rational and b is irrational

A

a+b irrational
if a does not =0, ab irrational

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10
Q

if x does not = 1, then x + x^2 + x^3 +…+ x^n=

A

x(1-x^n) / 1-x

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11
Q

if -1<x<1, then x + x^2 + x^3 +… =

A

x / 1-x

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12
Q

if -1< x <1, then x + x^2 + x^3 +…=

A

1 / 1-x

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13
Q

every real number has decimal expression of form

A

x=a0.a1a2a3…

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14
Q

rational form of decimal steps

A

express as fractions over a power of 10
take out common factor to leave expression for geometric series

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15
Q

inequalities rules

A
  1. if x is a real number either x>0 or x<0 or x=0 and just one of these is true
  2. if x>y then -x<-y
  3. if x>y and c is a real number then x+c>y+c
  4. if x>0, y>0 then xy>0
  5. if x>y, y>z then x>z
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16
Q

how to find x in equations with modulus steps

A
  1. where does the modulus change?
  2. list cases for between each change
  3. case for both +ve, one +ve and one -ve and both -ve - evaluate inequality for each
  4. reach overall range based on cases
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17
Q

triangle inequality

A

|x+y| </= |x| + |y|

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18
Q

dividing complex numbers

A

multiply fraction by complex conjugate of denominator

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19
Q

how many complex roots does every quadratic have?

A

2

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20
Q

what is |z|

A

modulus of complex number

distance to origin

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21
Q

argument of complex number

A

angle between the positive x axis and line from z to the origin

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22
Q

polar form of a complex number

A

z=|z|(cosθ+isinθ)

adding multiples of 2pik does not change

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23
Q

principle argument

A

arg(z) between negative pi and positive pi

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24
Q

de moivre

A

z1z2=r1r2((cosθ1+θ2)+isin(θ1+θ2)

=r1r2e^i(θ1+θ2)

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25
Q

z^n=…

A

r^n(cosnθ+isinnθ)

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26
Q

z^-n

A

r^-n(cosnθ-isinnθ)

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27
Q

i theta form of complex numbers

A

z=re^iθ

e^iθ=e^i(θ+2kpi)

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28
Q

nth roots of unity

A

Zs that satisfy z^n=1

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29
Q

fundamental theorem of algebra

A

every polynomial equation of degree at least 1 has a root in C

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30
Q

polynomial of degree n factors into…

A

n linear equations

n roots in C

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31
Q

every real polynomial factorisies as a product of…

A

real linear and real quadratic polynomials

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32
Q

the non-real roots always…

A

come in complex conjugate pairs

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33
Q

(x-alpha)(x-beta)=

A

x^2-(alpha+beta)x + alphabeta

alpha+beta=-a
alphabeta=b

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34
Q

(x-alpha)(x-beta)(x-gamma)=

A

x^3+ax^2+bx+c

alpha+beta+gamma=-a
alphabeta+alphagamma+betagamma=b

alphabetagamma=c

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35
Q

p(n) true for all positive integers if

A
  1. p(n) true
  2. for all n,if p(n) true, p(n+1) also true

assume p(n) true, consider p(n+1)

36
Q

PMI II

(don’t need to start at 1)

A
  1. p(k) true
  2. for all n>/=k, p(n) true, p(n+1) true, then p(n) is true for all integers n>/=k
37
Q

strong mathematical induction

A
  1. p(k) true
  2. for all n, if p(k), p(k+1),…p(n) are true, p(n+1) true
  3. p(n) true for all n >/=k
38
Q

recurrence relation

A

terms defined as function of previous term

39
Q

convex

A

connect two points on surface together, line is contained inside polyhedron

platonic solids are all convex

40
Q

euler’s formula

A

V-E+F=2

for connected plane graph
v-e+f=1

41
Q

plane graph

A

collection of points, edges joining points.
no two edges cross

42
Q

connected plane graph

A

can go from any vertex to any vertex

43
Q

drawing plane graphs for platonic solids

A

imagine looking through a face

44
Q

regular polygon

A

all sides equal lengths, internal angels equal

faces are all same and each vertex belongs to the same number of edges

45
Q

platonic solids

A

tetrahedron
cube
octahedron
dodecahedron
icosahedron

46
Q

faces of platonic solids

A

cube - squares
tetrahedron, octahedron and icosahedron - triangles
dodecahedron - pentagon

47
Q

what are n and r for platonic solids?

A

n=number of sides on a face
r=number of edges each vertex belongs to

48
Q

what are the only regular convex polygons?

A

platonic solids

49
Q

only regular convex polygons are platonic solids proof

A

lemma: 2E=nF
lemma 2: 2E=rV
lemma 3: 1/r+1/n=1/2+1/E
lemma 4: either n=3 or r=3 comparing possibilities gives all the platonic solids

50
Q

quotient and remainder

A

b=quotient a + remainder

51
Q

if d divides a and d divides b then

A

d divides ma+nb

52
Q

coprime

A

hcf(a,b)=1

53
Q

if a and c are coprime and c divides ab, then

A

c divides b

54
Q

if p is prime and p divides ab then

A

either p divides a or p divides b

55
Q

fundamental theorem of arithmetic

A

integers greater than or equal to 2 are a product of primes and this prime factorisation is unique

56
Q

hcf(a,b) using prime factorisation

A

p1^min(a1,b1)p2min(a2,b2)

57
Q

lcm(a,b) using prime factorisation

A

p1^max(a1,b1)p2max(a2,b2)

ab/hcf(a,b)

58
Q

root n rational if and only if

A

n=m^2 for some m

59
Q

if a and b are coprime and ab a square

A

both a and b squares

60
Q

a and b coprime and ab is an nth power for some n (natural number),

A

both a and b also nth powers

61
Q

diophantinc equation

A

polynomial in two or more variables in which only integer solutions considered

eg: x^3=y^2

62
Q

how to approach diophantinc equations

A

does euclid help?
can you factor?
can you apply FTA?

63
Q

how to check if a number is prime

A

check if p divides n for all prime numbers from 2 to root n

if not, n is prime

64
Q

let pi(n) be the number of primes up to n

ratio of pi(n) and n/loge(n)…

A

tends to 1 as n tends to infinity

65
Q

every integer greater than 2 is…

A

the sum of two prime numbers

66
Q

every odd integer greater than 5 is…

A

the sum of three prime numbers

67
Q

if m divides b-a

A

a congruent to b mod m

68
Q

suppose a≡b(mod m) and c≡d(mod m)

A

a+c≡b+d (mod m)
ac≡bd (mod m)

69
Q

if a≡b(mod m) and n +ve

A

a^n≡b^n(mod m)

70
Q

method of successive squaring

A

keep squaring
make power up using indices
multiply congruences

71
Q

rule of nines

A

n divisible by 9 if sum of digits divisible by 9

72
Q

a and m coprime, x,y integers then

A

xa≡ya(mod m)

x≡y(mod m)

73
Q

form of congruence equations

A

ax≡b(mod m)

74
Q

condition for congruence equation to have solution

A

if and only if hcf(a,m) divides b

75
Q

finding solutions for congruence equations

A

kd=b
d=sa+tm
b=kd=k(sa+tm)
aks=b-k+m≡b(mod m)

setting x=ks gives solution

76
Q

Zm=

A

{0,1,…,m-1}

77
Q

adding in Zm

A

a+b=a+b(mod m)

78
Q

multiplying in Zm

A

a.b=a.b(mod m)

79
Q

Zm for low numbers

A

can check all possibilities using table

(sub different x values into expression and then evaluate in modulo m)

80
Q

Fermat’s little theorem

A

a not divisible by p

a^p-1≡1(mod p)

can use to check if a number is prime

81
Q

if p and q are distinct primes, a an integer not divisible by p or q

A

a^(p-1)(q-1)≡1(mod pq)

82
Q

for x^k≡b(mod m)

A

solution x are called kth roots of b modulo m

looking for x such that x^k≡b in Zm

83
Q

if k is coprime to p-1

A
  1. positive integer s such that sk≡1(mod p-1)
  2. for any b not divisible by p
    x^k≡b(mod p)

has unique solution for x modulo p

solution: x≡b^s(mod p)

84
Q

RSA encoding setting up

A

two large prime p and q

  1. multiply so N=pq
  2. find m=(p-1)(q-1)
  3. find e coprume to M

whole message broken into chunks

85
Q

RSA encoding for each segment x

A

compute y≡x^e(modN)
using successive squaring

send y value and repeat

86
Q

RSA decoding

A

find d such that de≡1(mod(p-1)(q-1))

use Euclid to find d such that 1=ed+(p-1)(q-1)t

let x=y^d(mod pq) and use successive squaring

Maths 1 (5 decks)

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