SUMS & PRODUCTS
summation Operator # Sigma
I
✗i = ✗ mt ✗mint . . .
+ ✗ + ✗n
n,
Em
→ Sum for i is m ton of xi
Properties
✗ it ✗I =
it i #ADD/THEY
✗
i
= C i # HOMOGENEITY
-
n "" him
)- ⇐,
✗i
F)
-
i=nµ
✗i
=] -
⇐,
✗i #EXTENDING TERMS
F- I ,
c = he # CONSTANT TERM
-
n
)- ( ✗i -
✗
i
- 1) =
✗n
-
✗
o # TELESCOPIC SERIES
Example :
F-[ ✗ = A + 2+3
, Product Operator #Pi
✗ ✗ ✗n
✗i =
Xm ✗ ✗ ✗ ✗
. - -
my n→
Em
→ Product for i is m ton of ✗i
Properties
=
"" "+m
,
✗i ✗
µ i=nµ+i # =
⇐,
✗i # EXTENDING TERMS
IT ⇐,
<✗ = C
"
⑦¥ ,
✗i #CONSTANT TERM
Example :
3
✗ = 1×2×3
Combination Examples :
% [
-
)-n=i ✗
✗ = (1) + (1×2) + (1×2×3)
1/-3 )
-
n "
n=1
-
X -
-
✗
A
=] -
✗ h=1
✗ =
(1) ✗ (1+2) ✗ (1+2+3)
?⃝
?⃝
?⃝
,SETS.FI/NUToNS&EQ4ATioNS
Sets
Collection of Elements
S={ en.eu } . . .
→ ees
special Sets
empty P prime numbers
IN natural numbers I
integers
④ rational numbers E
integers (even)
IR real numbers ① integers ( odd)
Intervals
(× , y) open ( x ,y excluded) also IX. ✗ I
IX. ✗ I closed ( ✗ ✗ included)
IX. Y ) semi-open also IX. YI
Conditions
{✗ E5A } also { ✗ ES1A}
→ All ✗ ins satisfy Condition A
{ ✗ ES1A }
→ All ✗ ins except ✗ in A
, Operations
Example : A =
{1,2413--12,3}
A UB union ( AuB={12,3 })
AAB intersection ( An B={2})
A \ B difference (A) 13=1^13 })
subsets & Super sets
Example : A={12,3} B={2,3 } G- {3,4 }
, ,
Be A → B is a Subset of A
C ¢ A → C is not a Subset of A
A3B → A is a Superset of B
A ✗ C → A is not a Superset of C
Functions
of
Assignation a
unique number in domain D
for each value of variable ✗ ( argument
✗ → ffx)
✗
= f- A)
summation Operator # Sigma
I
✗i = ✗ mt ✗mint . . .
+ ✗ + ✗n
n,
Em
→ Sum for i is m ton of xi
Properties
✗ it ✗I =
it i #ADD/THEY
✗
i
= C i # HOMOGENEITY
-
n "" him
)- ⇐,
✗i
F)
-
i=nµ
✗i
=] -
⇐,
✗i #EXTENDING TERMS
F- I ,
c = he # CONSTANT TERM
-
n
)- ( ✗i -
✗
i
- 1) =
✗n
-
✗
o # TELESCOPIC SERIES
Example :
F-[ ✗ = A + 2+3
, Product Operator #Pi
✗ ✗ ✗n
✗i =
Xm ✗ ✗ ✗ ✗
. - -
my n→
Em
→ Product for i is m ton of ✗i
Properties
=
"" "+m
,
✗i ✗
µ i=nµ+i # =
⇐,
✗i # EXTENDING TERMS
IT ⇐,
<✗ = C
"
⑦¥ ,
✗i #CONSTANT TERM
Example :
3
✗ = 1×2×3
Combination Examples :
% [
-
)-n=i ✗
✗ = (1) + (1×2) + (1×2×3)
1/-3 )
-
n "
n=1
-
X -
-
✗
A
=] -
✗ h=1
✗ =
(1) ✗ (1+2) ✗ (1+2+3)
?⃝
?⃝
?⃝
,SETS.FI/NUToNS&EQ4ATioNS
Sets
Collection of Elements
S={ en.eu } . . .
→ ees
special Sets
empty P prime numbers
IN natural numbers I
integers
④ rational numbers E
integers (even)
IR real numbers ① integers ( odd)
Intervals
(× , y) open ( x ,y excluded) also IX. ✗ I
IX. ✗ I closed ( ✗ ✗ included)
IX. Y ) semi-open also IX. YI
Conditions
{✗ E5A } also { ✗ ES1A}
→ All ✗ ins satisfy Condition A
{ ✗ ES1A }
→ All ✗ ins except ✗ in A
, Operations
Example : A =
{1,2413--12,3}
A UB union ( AuB={12,3 })
AAB intersection ( An B={2})
A \ B difference (A) 13=1^13 })
subsets & Super sets
Example : A={12,3} B={2,3 } G- {3,4 }
, ,
Be A → B is a Subset of A
C ¢ A → C is not a Subset of A
A3B → A is a Superset of B
A ✗ C → A is not a Superset of C
Functions
of
Assignation a
unique number in domain D
for each value of variable ✗ ( argument
✗ → ffx)
✗
= f- A)
