Algebra
Set Theory:
Set – is a gathering together into a whole of definite, distinct objects of our perception or of our thought
– which are called elements of the set.
Cardinality of a set – the number of members of “S” usually denoted as ¿ S∨¿or n ( S ) .
Special Sets
1. {} or ∅ - Null Set or Empty Set.
2. { x } or x – Unit Set, which contains exactly one element.
3. P – The set of all prime numbers.
4. Z – The set of all integers.
5. R – The set of all real numbers.
6. H – The set of all quaternions.
7. N – The set of all natural numbers.
8. Q – The set of all rational numbers.
9. C – The set of all complex numbers.
Subsets – If every element of A is contained in B, then A ⊆ B.
Proper Subset – When A ⊆ B and A ≠ B , then A is a proper subset to B.
Fig. 1 - A ⊆ B
Complement (Absolute Complement) – Denoted as A' ∨ A c where all elements do not belong to A .
Fig. 2 – Visual representation of Absolute Complement
Relative Complement – Denoted as A ¿ where all elements that belong to A but di not belong to B.
Fig. 3 – Visual representation of Relative Complement
Union – The union of A and B denoted as A ∪ B is the set of all things that are members of A and B.
Fig. 4 – Visual representation of a Union of two set
Intersection – The intersection of A and B denoted by A ∩ B is the set of all things that are members of
A and B.
Fig. 5 – Visual representation of Intersection of two set
Principle of Inclusion and Exclusion:
Provides an organized method to find the number of elements in the union of a given group of sets, the
size of all possible interactions among the sets.
¿ i=1 ¿ n A 1=∑ | A 1|−∑ | A i ∩ A j|+ ∑ | A i ∩ A j ∩ Ak|−…+(−1) | Ai ∩… ∩ A n|
n−1
Worded Problem:
Number Problem:
Clock Problem:
Angle travelled by the Minute Hand in t mins.
θm =θo +6 t
Angle travelled by the Hour Hand in t mins.
θm =θo +0.5 t
NOTE: All angles are in degrees.
Money Problem:
Interest:
I =Cost × Rate
Profit:
P=Selling Price−Cost
Discount:
D−Cost × [ 1−Rate ]
Motion Problem:
Distance Formula:
d=Vt
Speed Relation:
Given an object with V 1 with an environmental factor that affects velocity namely V 2 the
speed relation between the two object is:
V =V 1+V 2 V =V 1−V 2
Alternatively, given a value of V 3 where V 3 is the object factor in relativistic terms between
object, the velocity is as follows:
V =V 1−V 2 V =V 1+V 2
Work Problem:
Work
Rate=
Time
Mixture Problem:
a1 a
a2
+¿ ¿
b1 b
b2
x1 x2 x 1+ x2
a 1 x 1+ a2 x2= ( x1 + x 2 ) % of a
or
b 1 x 1+ b2 x 2=( x1 + x 2 ) % of b
Variation Problem:
Direct Variation:
y=kx
Inverse Variation:
k
y=
x
Joint Variation:
y=kxz
Combined Variation:
kz
y=
x
Sequences and Series:
Arithmetic Progression:
a n=a1 + ( n−1 ) d
Sum of Arithmetic Progression:
n
Sn= ( a1+ an )
2
or
n
Sn =
2
( 2 a1 + ( n−1 ) d )
Arithmetic Mean:
n
1
A= ∑a
n i=1 i
Geometric Progression:
n−1
a n=a1 r
Sum of Geometric Progression:
a 1 ( 1−r n )
Sn =
1−r
Sum of Infinite Geometric Progression (|r|<1) :
a1
Sn =
1−r
Geometric Mean:
(∏ )
n 1
x i = √n x 1 x 2 … xn
n
i=1
Harmonic Progression:
1 1 1
, ,…,
a1 a2 an
Harmonic Mean:
n
H=
1 1 1
+ +…+
a1 a 2 an
