MATH1131 Lecture Notes - Lecture 9: Asteroid Family, Implementation Force
3 Jun 2018
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2 Complex Numbers
2.L Sets
A set is an unordered collection of elements. Repeatcd
elements are discarded. For example consider the set of
1 rrnd 2:
{t,2) : {2,I1 : {2,2,2,1, 1, 1, 1, 1, 1}.
There are two styles for rvriting a set:
o sets can have all their eiemcnts placed between "{"
anci ")" as above.
o sets can be spccified by two criteria seperatcd bv
a : or I symbol. The left critcria tells you where
the elements of this set come from. The riglrt crite'
ria gives extra conditions tlie elements must satisfv.
For e.xarnolet ur u^arrrPl. it114u *4h $l! yfttrl.1
{neZ:1 <n<2}
says that the elements n belong to the integers (see
lielow). In addition they nlrst satisfl'the additiorral
criteria of being between one and two.
i
o C - "subset of". if Sr and 52 are two sets and I
rvrite Si C 52 thel this means "ctery element iu
.91 is also in S2". Il -vou use C instead, then y6rl
insist that fil Sz.
\, Similarlv for I and l,
o O - "intersection of". Think "AND". So 51 i 52 is
' a set consisting of ali elernents that are in 51 AND
S2
o U - "union of" . Think "OR". So ,Sr U ,Sz is a set
consisting of all elements ihat are in either 51 OR
52 (or both).
So we ha,ve
S c &*g.'&c @rie mt
Herc are the rnost of the sets vou wili encounter:
. 0 : {} The empty set. It contains no elements.
. N - {0, 1,2,3,.. .} called the natural numbers.
o Z: {...,-2,-1,0, 1,2,3,...} called the inte-
gers. i
I LL*
o Z+ : {1,2,3. . . .} ca11ed the positive integers.
. Q: {i , o qtZ. forq / 0} called the ratie
nal numbers. These are just all the fr"actions.
(Can you rvrite them all out in a line. like we did
with N[ and 27)
o IR :? calletl the real numbers. It contains all
the rational nurr*;ers plrrs lots more. Numbers such
as t/l,r,logr7, sin 33" are all real. (Can y.ou write
them all in a line?)
Here is some notation for working with sets:
. € "elenreut of". If S is a set, and I write e € .9
tlior this lncans "u is in.side S".
. -a-i
2
Example 2.1. Simplifi,
o {r elnl 1r2 :-z)(r + 1) : o} : I tfi , --n , .ll
o {a; € Ql (r'- 2)(r + i) : oi - [ -'l
o {r € zl (r2 -z)(r+ 1) : ol : t"l
r {r e Nl (r2 ;2)(r + 1) :0} : [ 1-- (
Q . [. eR fi tx'+z) (x+r) = oJ . [rJ
$, t/ f,
