,At :
Natural numbers [1 .
2 3
.
...
3
: Integers 2-2 .
-1
.
0
.
1
.
2 ...
3
⑫: Rational numbers [IR where b 03
: Real
Q :
Complex
Set Builder Notation
y
= & set with objects I rule3
eg. k = GoEIR/ocEN & 3 x 7753 x =
.
1x
Cardinality =
&
number of elements in a set
① =
empty set
Set : well-defined mathematical collection of distinct objects
The Union (U)
AUB = GolocEA or sEB
1 .
AUB =
BUA :.
Commutative
.
2 AU (BUC) = (AUB)UC :. Associative can reorder brackets)
The Intersection (1)
Al B =
ExloEA and of BY
1 . A 1B =
B1A : commutative
2AM (BMC) =
(AMB) 1C : Associative
The Set Difference (1)
AlB = B)A
AlB =
GxlocfA and xEB3
, in s is in set A
every element
Subsets (f)
A can = S
~
&
A set S is said to be a subset of a Set A (SEA) ,
it each element of S is an element of A
.
of A is in C but differing
Proper Subset (C) Every element C has at least one element
-Every element of set A is in set c and A C
&: not a subset
is subset for all sets
Empty set
always a
# c c QC IR
Intervals
C ,
0
,
< = not
including
C .0, including
Inequalities
I . 18 add and bCC : a C 4 . 18 a < b and C>0 :. ac LbC
.
2 18a(b : a + c >b + c 5.
5 18a < b and <O :. ac > be
.
3 Ifa < b and c (d : a + c b + d 6 .
18 09 < b :)
Proof that any function =
sum of odd and even function
g(x) f(x)
=
+ f) -
x)
n (x) = C(x) -
Gl -
x)
g(x) + h(x) = 2f(x)
Egx) + zh(x) =
f(x)
, Functions
A rule which maps you from elements of one set of elements to elements of another
8 :
A >
-
B
↓
For an element ofA there is a single element ScodEB
Domain :
the set of all values which the function can be applied to
Codomain : the set of all possible values that the function could take
Range
:
the set of all possible values that the function actually takes when applied to the elements
in the domain .
can be represented as 11) Words &I Formula 13/Graph (4) Table
Piecewise function
multiple pieces
function defined different intervals
by multiple sub-functions in
~
A .
I rule
Do each 11 as + 10
Gimme
1.
- G
eg So =
. .
2 Find the if .
3 . Find which equation fits
↑
function)
&
10 3 is not apart of 4
Simplify
30
.
.
10 0 is a part of the function
1z'
.
⑧ Not apart
· Apart
-
7
-
y = x
-
Absolute value function y
= x
-x)x(0)
(x) =
E I
-x)x20
eg .
(4) = 44 and 1-3) =
3
Natural numbers [1 .
2 3
.
...
3
: Integers 2-2 .
-1
.
0
.
1
.
2 ...
3
⑫: Rational numbers [IR where b 03
: Real
Q :
Complex
Set Builder Notation
y
= & set with objects I rule3
eg. k = GoEIR/ocEN & 3 x 7753 x =
.
1x
Cardinality =
&
number of elements in a set
① =
empty set
Set : well-defined mathematical collection of distinct objects
The Union (U)
AUB = GolocEA or sEB
1 .
AUB =
BUA :.
Commutative
.
2 AU (BUC) = (AUB)UC :. Associative can reorder brackets)
The Intersection (1)
Al B =
ExloEA and of BY
1 . A 1B =
B1A : commutative
2AM (BMC) =
(AMB) 1C : Associative
The Set Difference (1)
AlB = B)A
AlB =
GxlocfA and xEB3
, in s is in set A
every element
Subsets (f)
A can = S
~
&
A set S is said to be a subset of a Set A (SEA) ,
it each element of S is an element of A
.
of A is in C but differing
Proper Subset (C) Every element C has at least one element
-Every element of set A is in set c and A C
&: not a subset
is subset for all sets
Empty set
always a
# c c QC IR
Intervals
C ,
0
,
< = not
including
C .0, including
Inequalities
I . 18 add and bCC : a C 4 . 18 a < b and C>0 :. ac LbC
.
2 18a(b : a + c >b + c 5.
5 18a < b and <O :. ac > be
.
3 Ifa < b and c (d : a + c b + d 6 .
18 09 < b :)
Proof that any function =
sum of odd and even function
g(x) f(x)
=
+ f) -
x)
n (x) = C(x) -
Gl -
x)
g(x) + h(x) = 2f(x)
Egx) + zh(x) =
f(x)
, Functions
A rule which maps you from elements of one set of elements to elements of another
8 :
A >
-
B
↓
For an element ofA there is a single element ScodEB
Domain :
the set of all values which the function can be applied to
Codomain : the set of all possible values that the function could take
Range
:
the set of all possible values that the function actually takes when applied to the elements
in the domain .
can be represented as 11) Words &I Formula 13/Graph (4) Table
Piecewise function
multiple pieces
function defined different intervals
by multiple sub-functions in
~
A .
I rule
Do each 11 as + 10
Gimme
1.
- G
eg So =
. .
2 Find the if .
3 . Find which equation fits
↑
function)
&
10 3 is not apart of 4
Simplify
30
.
.
10 0 is a part of the function
1z'
.
⑧ Not apart
· Apart
-
7
-
y = x
-
Absolute value function y
= x
-x)x(0)
(x) =
E I
-x)x20
eg .
(4) = 44 and 1-3) =
3
