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if A and B are sets
and all elements of A are elements of B
(all A's are B's)
.
Given two sets A and B,
we
A=B
if
ACB and BCA
(A-B means that A and B
are
we
Say
that if XEA, then x GB
dif names
for the same collection (set)
→ abbrev: A CB
Bis a superset of A
We
say:
:DA is a subset of
B
when we
we say:
We
aren't
All A's
are
B's
sure that "All B's are A's
A&B ACB, but at least one
→
•
element of B isn't A.
ex) Wthe set of whole numbers
R- the sex
of rational numbers
"
Why different names?
What's the purpose?.
ex) A→ A
B→A
As we
we prove
=A
•
Picture
We use
3
ACB and BCA
> A=B.
is worth 1000 words."
-pictures" for viewing" sets,
Lircle/venn diagrams
closed curves.
we draw A&B:
BUT
whole number, for ex., ½/2.
R is
integer
So
whole number
is a
integer
rational number : n=
every
n = ½
not
evey
Rational number is a
we draw ACB:
B
(A)
WER
WER (they've both sets)
Let's
sary
A Frenchman
Bmusician
A B
→
BA
Take a look at:
<
u, b
unequal
if are dif. numbers
it's either a<b
12
2:C" and "<"
or
a>b.
( some similarities, love norize the lif.)
B's
which aren't As
"clotted line "Is it true that
some B's are not A's.?"
It's possible that A¢B und B¢A.
DRAW
AA's which
are not B
B's which
aren't Ai
Elements which.
are both A's and B's
B

ページ4:

we know ACA
A & A but ACA.
C. 2 special sets
A SET MIGHT have No members.
ex) Analytic Geometry: x=!
9
(a)
↓
WHY?
For
any
set,
there's THE TEST FOR
I: x-axis
MEMBERSHIP, so it's possible that nothing graph a point
Survives.
given
number ">5" and <3
ex) Set any
no number survive
→test: objective and well-defined
Curious)
(b)
xy-plane
3D
space
a straight line
a plane
When we draw Venn diagrams,
We make Sure
every
set is drawn
within I (a rectangle)
THE TEST than the elements.
• In MATH,
we want to know more
about)
.
The
Set:
Ø (not 0)
empty
(the null set)
→the number
A
of elements in p is O. D. Set- builder
set: all numbers both even and odd>
•The universal set: I
ex)
(ehe universe of discourse)
: for any element b
beI and
15
see A
ACI
A and I serve as upper/lower bounds
in sets:
for each set A,
it's true that CACI
in 9th grade, we can't
factor x+1 (we only consider PR)
BUT in 11th grade, we can..
x²+/= (x+1)(x-1) (we consider C)
if the universe of discourse werk,
then x²+1 can't be factored.
but if the universe of discourse were C,.
then it could be factored.
.
(1) Roster
VS.
Roster method
Method (explicitly)
A = {1, 2, 3, 4, 5, 528, 9} ✓.
A = 1, 2, 3, 4, 5, 6, 7, 8, 9 " X
equality: contain same members,
but does not have
same order.
to
be in the
.
ex) A= {12, 3, 4, 5} = { 3, 1, 2, 5, 4}
• We
•
never
list the same element
more than once..
A = { 1, 1, 2, 2, 3, 3} "",
A={1,2,3}
(exception
we want
to
we consider the orders only when
investigate
the various
arrangement for the word "Mississippi")

ページ5:

.
(2) set-builder notation (implicitly)
ex) {x: xεR}
ex) A people alive on 1929/1/1
ex) Sometimes
We say
are the same.
We actually
mean:
x=4x+3 and (6x-1) (x-3)
B→
"
"
17
1960/12/12
{x: x εA and x {B}
Note: notice the compactness
the set-builder method
emphasizes its test for membership
The strengths and weaknesses of the
2
methods.
(1) Roster method.
Strength
: we know immediately what're the
elements (easiest possible test for membership).
weakness: we can't lije infinite items.
Some ppl write: 1,2,3,
2 different equations have the same
Set.
S= {x: x =4x+3=0}
T= {x: (x-3) (x-1)=0}
•
S= T = {1,3}
2 descriptions are
•
actually equilavent
•
•
•
•
.
•
•
Solution
•
•
•
.
vague, subjective
•
We
can't
lise
many
items
too.
•
in
hard to list
(b) strength for set-builder method.
ex) Find the
modern
math
roots for x = -4x+3=0
(2-1)(2-3)=0 → x=1 or 3.
Find the solution set,
x²-4x+3=0
S, for the equation
S = {1,3} (roster method)
S= {x: x = 4x+3=0} (Sex-builder)
Algebra: turn the [set-builder } into the {roster}
• These 2 sets are equal".
Let's talk about the equivalence of 2 or
more
equations.
•
•