Complete Mathematics Summary with Definitions & Explanations

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)