Propositional and Mathematical Statements:
Connectives and Truth Tables:
Connectives such as AND (∧), OR (∨), NOT (¬), IMPLIES (→), and IF AND
ONLY IF (↔) modify and combine propositional statements. They are
represented in truth tables:
(P) (Q) (P∧Q) (P∨Q) (¬P) (P→Q) (P↔Q)
T
T
T
T
F
T
T
T
F
F
T
F
F
F
F
T
F
T
T
T
F
F
F
F
F
T
T
T
1. Inverse: If the original statement is "if P then Q" (symbolically
represented as P → Q), then the inverse is "if not P then not Q"
(symbolically represented as ~P → ~Q). So, we just negate both the
hypothesis and the conclusion.
2. Converse: For the original statement "if P then Q" (P → Q), the
converse is "if Q then P" (Q → P). So, the hypothesis and conclusion
are being swapped.
3. Contrapositive: The contrapositive of "if P then Q" (P → Q) is "if not Q
then not P" (~Q → ~P). So, it's when you negate both the hypothesis
and the conclusion and then swap them.
It's worth noting that an implication and its contrapositive always have the
same truth value (if one is true, so is the other; if one is false, so is the
other). However, an implication and its converse or an implication and its
inverse do not necessarily share the same truth value.
Rules of Logical Equivalence
Identity laws:
P AND T ≡ P: (P
P OR F ≡ P: (P
∧ T) ≡ P
∨ F) ≡ P
Domination laws:
P AND F ≡ F: (P
P OR T ≡ T: (P
Negation laws:
∧ F) ≡ F
∨ T) ≡ T
P OR ~P ≡ T: (P
∨ ¬P ) ≡ T
P AND ~P ≡ F (P
∧ ¬P ) ≡ F
Double Negation Law:
P ≡ P: (¬¬P ) ≡ P
¬¬
Idempotent laws:
P AND P ≡ P: (P
P OR P ≡ P: (P
∧ P) ≡ P
∨ P) ≡ P
Commutative laws:
P AND Q ≡ Q AND P: (P
P OR Q ≡ Q OR P: (P
∧ Q) ≡ (Q ∧ P )
∨ Q) ≡ (Q ∨ P )
Associative laws:
P AND (Q AND R) ≡ (P AND Q) AND R: (P
P OR (Q OR R) ≡ (P OR Q) OR R: (P
∧ (Q ∧ R)) ≡ ((P ∧ Q) ∧ R)
∨ (Q ∨ R)) ≡ ((P ∨ Q) ∨ R)
Distributive laws:
P AND (Q OR R) ≡ (P AND Q) OR (P AND R):
(P ∧ (Q ∨ R)) ≡ ((P ∧ Q) ∨ (P ∧ R))
P OR (Q AND R) ≡ (P OR Q) AND (P OR R):
(P ∨ (Q ∧ R)) ≡ ((P ∨ Q) ∧ (P ∨ R))
De Morgan’s laws:
NOT (P AND Q) ≡ (~P OR ~Q): (¬(P
∧ Q)) ≡ (¬P ∨ ¬Q)
NOT (P OR Q) ≡ (~P AND ~Q): (¬(P
∨ Q)) ≡ (¬P ∧ ¬Q)
Absorption laws:
- P OR (P AND Q) ≡ P: (P
- P AND (P OR Q) ≡ P: (P
Material Implication
P → Q ≡ ¬P v Q
Rules of Inference:
Modus Ponens:
P→Q
P
⊢ Q
∨ (P ∧ Q)) ≡ P
∧ (P ∨ Q)) ≡ P
Modus Tollens:
P→Q
¬ Q
⊢ ¬P
Hypothetical Syllogism:
P→Q
Q→R
⊢ P → R
Disjunctive Syllogism:
P∨Q
¬ P
⊢ Q
Addition:
P
⊢ P ∨ Q
Simplification:
P∧Q
⊢ P
and
P∧Q
⊢ Q
Conjunction: (P , Q ⊢ P ∧ Q )
Resolution: ( (P ∨ Q), (¬ P ∨ R) ⊢ (Q ∨ R) )
Universal Generalization: If ( P(a) ) holds for an arbitrary element ( a )
in the domain, then ∀xP(x) holds.
Predicates and Quantifiers:
Predicates:
Predicates are functions that take one or more variables and produce a truth
value. For instance, ( P(x) ) might denote "x is an even number."
Quantifiers:
Universal Quantifier (∀):
Denotes "for all" or "for every." For example, ∀ x (x > 0) means "for all x,
x is greater than 0."
Existential Quantifier (∃):
Denotes "there exists." For instance, ( ∃ x (x > 5) ) means "there exists
an x greater than 5."
Theorems:
Universal Instantiation: From ( ∀ x P(x) ), deduce ( P(c) ) for a
particular element ( c ).
Existential Instantiation: From ∃ x P(x), deduce P(c) for a chosen
element ( c ).
Universal Generalization: If ( P(a) ) holds for an arbitrary element (a)
in the domain, then ∀ x P(x) holds.
Set Theory:
Set Identities:
Commutative Laws: ( A ∪ B = B ∪ A ) and ( A ∩ B = B ∩ A )
Associative Laws: ( (A ∪ B) ∪ C = A ∪ (B ∪ C) ) and ( (A ∩ B) ∩ C = A
∩ (B ∩ C) )
Distributive Laws: ( A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) ) and ( A ∩ (B ∪ C)
= (A ∩ B) ∪ (A ∩ C) )
Identity Laws: ( A ∪ ∅ = A ) and ( A ∩ U = A )
Complement Laws: ( A ∪ A' = U ) and ( A ∩ A' = ∅ )
Relations:
Relation Properties:
Reflexivity: ∀ x (x, x) belongs to the relation.
Symmetry: If (a, b) belongs to the relation, then (b, a) also belongs.
Transitivity: If (a, b) and (b, c) belong to the relation, then (a, c) also
belongs.
Equivalence Relation: A relation that is reflexive, symmetric, and
transitive.
Functions:
Functions map elements from one set (domain) to another (codomain).
Different types of functions include:
Injective (One-to-One): Each element of the domain maps to a distinct
element in the codomain.
Surjective (Onto): Every element in the codomain has at least one
corresponding element in the domain.
Bijective: Combines the properties of injective and surjective functions.
Summation and Its Different Formulas
Summation (Σ) is a mathematical operation that represents the addition of a
sequence of numbers or terms. It's symbolized by the Greek letter sigma ((Σ
).
Basic Summation Notation:
The summation notation represents the sum of terms from (i = m) to (n) of a
given expression (f(i)):
Σ
n
i=m
f (i)
Here are some key summation formulas and properties:
Arithmetic Series:
The sum of an arithmetic series (a , a , a , … , a
difference (d), and first term (a ) is given by:
1
2
3
n)
with (n) terms, common
1
Sn =
n
2
n
⋅ (a 1 + a n ) =
2
⋅ (2a 1 + (n − 1)d)
Geometric Series:
The sum of a geometric series (a , a , a , …, a ) with (n) terms, first term (
a ), and common ratio (r) is calculated as:
1
2
3
n
1
Sn =
a 1 ⋅(r
n
−1)
r−1
f or
r ≠ 1
Summation Properties:
Linearity: (Σ
n
i=m
(c ⋅ f(i) + g(i)) = c ⋅ Σ
n
i=m
f (i)
+Σ
n
i=m
g(i))
Changing the Index of Summation: Shifting indices in a sum does not
affect the result.
Splitting a Sum: ( Σ
n
i=m
f(i) = ∑
k
i=m
f(i) + Σ
n
i=k+1
f(i))
Telescoping Sums: Sum cancellation where many terms cancel each
other out, leaving only a few remaining terms.
Arithmetic Mean of Summation: (
of the terms.
1
Σ
n
n
i=1
f(i) ) represents the average
Double and Triple Summations:
Double (∑∑) and triple (∑∑∑) summations involve nested sums. For
example:
Double Summation: (∑
Triple Summation: (∑
Arithmetic:
m
i=1
m
i=1
∑
n
n
j=1
)
f (i, j)
∑ j=1 ∑
p
k=1
f(i, j, k))
GCD (Greatest Common Divisor) and LCM (Least
Common Multiple):
GCD: gcd(a, b) represents the largest integer that divides both (a) and
(b) without leaving a remainder.
LCM: lcm(a, b) denotes the smallest multiple that is divisible by both (a)
and (b).
For example, the GCD of 8 and 12 can be found by considering their prime
factorizations:
Factors of 8: (2^3)
Factors of 12: (2^2 \times 3)
Common factors: (2^2 = 4)
Greatest Common Divisor: (4)
Modulo Arithmetic:
Modulo arithmetic involves operations like (a \mod b), which returns the
remainder when (a) is divided by (b). For instance, (17 \mod 5 = 2) means
dividing 17 by 5 yields a remainder of 2.
Matrix Algebra
Zero Matrix
A zero matrix, denoted by O , is a matrix in which all the elements are
zero.
Example:
0
O = [
0
0
]
0
Transpose of a Matrix
The transpose of a matrix is obtained by interchanging its rows and
columns.
It is denoted by A^T , where A is the original matrix.
Example:
If
1
A = [
4
2
5
3
]
6
, then
⎢⎥
⎡
A
T
=
⎣
Symmetric Matrix
1
4
2
5
3
6
⎤
⎦
Original matrix is the same as the Transposed Matrix
A symmetric matrix is a square matrix that is equal to its transpose.
In other words, A = A^T .
Example:
1
A = [
2
A is symmetric because A = A^T.
Vector Dot Product
2
]
3
The dot product of two vectors is a scalar value representing the cosine
of the angle between them multiplied by their magnitudes.
Example:
If
1
⎡ ⎤
v =
2
⎣ ⎦
3
and
4
⎡ ⎤
u =
5
⎣ ⎦
6
, then
v ⋅ u = 1 × 4 + 2 × 5 + 3 × 6 = 32
.
Identity of a Matrix
The identity matrix, denoted by I , is a square matrix with ones (1) on
its main diagonal and zeros (0) elsewhere.
Example:
1
I = [
0
0
]
1
When multiplied by another matrix, the identity matrix leaves the matrix
unchanged.
Determinant of a Matrix
The determinant of a square matrix is a scalar value that can be
calculated from its elements.
It describes certain properties of the matrix and is denoted by det(A)
or |A| .
Example:
For a matrix
a
A = [
c
b
]
d
, the determinant is calculated as det(A) = ad - bc .
Matrix Operations
Matrix Addition
Matrix addition combines two matrices of the same order or dimension.
To add two matrices, add corresponding elements.
2x2 Matrix Example:
1
A = [
3
2
]
4
3
B = [
5
4
]
6
,
, then
1 + 3 = 4
4 + 2 = 6
3 + 5 = 8
4 + 6 = 10
A + B = [
]
4
A + B = [
8
6
]
10
3x3 Matrix Example
⎡
A =
⎣
1
2
3
4
5
6
7
8
9
⎤
⎦
,
⎢⎥
9
⎡
8
7
6
5
4
⎣
3
2
1
B =
,
A + B =
,
⎦
1 + 9 = 10
⎡
2 + 8 = 10
3 + 7 = 10
⎤
4 + 6 = 10
5 + 5 = 10
6 + 4 = 10
⎣
7 + 3 = 10
8 + 2 = 10
⎦
9 + 1 = 10
⎡
A + B =
⎣
Matrix Subtraction
⎤
10
10
10
10
10
10
10
10
10
⎤
⎦
Similar to matrix addition, but subtracts corresponding elements.
2x2 Matrix Example:
,
, then
1
A = [
3
2
]
4
3
B = [
5
4
]
6
1 − 3 = −2
2 − 4 = −2
3 − 5 = −2
4 − 6 = −2
A − B = [
]
2
A − B = [
−2
3x3 Matrix Example
2
3
4
5
6
7
8
9
9
⎡
8
7
6
5
4
⎣
3
2
1
A =
⎣
B =
]
−2
1
⎡
,
−2
⎤
⎦
⎤
⎦
,
⎢⎥
⎡
A − B =
⎣
,
1 − 9 = −8
2 − 8 = −6
3 − 7 = −4
4 − 6 = −2
5 − 5 = 0
6 − 4 = 2
7 − 3 = 4
8 − 2 = 6
9 − 1 = 8
−8
⎡
A − B =
−2
⎣
4
−6
−4
0
2
6
8
⎤
⎦
Scalar Multiplication
Multiplies a matrix by a scalar (single number).
2x2 Matrix Example:
1
A = [
3
,
2
]
4
k = 2
, then
1 ∗ 2 = 2
2 ∗ 2 = 4
3 ∗ 2 = 6
4 ∗ 2 = 8
2
kA = [
6
4
]
8
kA = [
]
3x3 Matrix Example
⎡
A =
⎣
,
1
2
3
4
5
6
7
8
9
⎤
⎦
k = 2
,
⎡
kA =
⎣
1 × 2 = 2
2 × 2 = 4
3 × 2 = 6
4 × 2 = 8
5 × 2 = 10
6 × 2 = 12
7 × 2 = 14
8 × 2 = 16
9 × 2 = 18
⎤
⎦
⎤
⎦
,
⎢⎥
⎡
kA =
2
4
6
8
10
12
16
⎦
18
⎣
14
Matrix Multiplication
⎤
Combines two matrices to produce a new matrix.
Number of columns in the 1st matrix must equal the number of rows in
the 2nd matrix for multiplication.
2x2 Matrix Example:
1
A = [
5
2
]
6
3
B = [
5
4
]
6
,
, then
1 × 3 + 2 × 5
AB = [
5 × 3 + 6 × 5
3x3 Matrix Example
1 × 4 + 2 × 6
13
] = [
5 × 4 + 6 × 6
33
1
2
3
4
5
6
7
8
9
9
⎡
8
7
6
5
4
⎣
3
2
1
⎡
A =
⎣
,
B =
,
⎡
AB =
⎣
16
]
40
⎤
⎦
⎤
⎦
1 × 9 + 2 × 6 + 3 × 3 = 30
1 × 8 + 2 × 5 + 3 × 2 = 24
1 × 7 + 2
4 × 9 + 5 × 6 + 6 × 3 = 84
4 × 8 + 5 × 5 + 6 × 2 = 69
4 × 7 + 5
7 × 9 + 8 × 6 + 9 × 3 = 138
7 × 8 + 8 × 5 + 9 × 2 = 114
7 × 7 + 8
,
⎡
AB =
⎣
30
24
18
⎤
84
69
54
138
114
⎦
90
Zero-One Matrix and its associated operations
Join and Meet
In lattice theory, a zero-one matrix represents a partially ordered set.
The join of two elements corresponds to matrix union, and meet
corresponds to matrix intersection.
These operations define relationships between elements in a partially
ordered set using zero-one matrices.