A Set is a well-defined collection of distinct objects or elements. Sets are usually denoted by capital letters and elements by lowercase letters.
Example:
Set A = {1, 2, 3, 4, 5}
Set B = {a, e, i, o, u}
A subset is a set where all elements are also contained in another set. If A ⊆ B, every element of A is in B.
Example:
A = {1, 2}, B = {1, 2, 3}
Here, A ⊆ B
Two sets are equal if they contain exactly the same elements, regardless of order.
Example:
A = {1, 2, 3}, B = {3, 2, 1}
Here, A = B
The universal set contains all elements under consideration, usually denoted by 'U'. Every other set is a subset of the universal set.
Example:
U = {1, 2, 3, 4, 5}, A = {2, 3}
Here, A ⊆ U
A set is finite if it contains a limited number of elements, and infinite if it has unlimited elements.
Example:
Finite Set: A = {1, 2, 3}
Infinite Set: N = {1, 2, 3, 4, 5, ...}
Set operations include union, intersection, and complement.
The union of two sets A and B (A ∪ B) includes all elements from both sets without repetition.
Example:
A = {1, 2, 3}, B = {3, 4, 5}
A ∪ B = {1, 2, 3, 4, 5}
The intersection of two sets A and B (A ∩ B) includes elements that are in both sets.
Example:
A = {1, 2, 3}, B = {2, 3, 4}
A ∩ B = {2, 3}
The complement of a set A (denoted A') includes all elements in the universal set U that are not in A.
Example:
U = {1, 2, 3, 4, 5}, A = {1, 2}
A' = {3, 4, 5}
The Cartesian product of two sets A and B (A × B) is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
Example:
A = {1, 2}, B = {x, y}
A × B = {(1, x), (1, y), (2, x), (2, y)}
The cardinality of a set is the number of elements in the set.
Example:
A = {5, 6, 7}
Cardinality of A = 3
Set theory is used in probability, database queries, and solving real-world logic problems.
Example:
Let A = students who play cricket, B = students who play football.
If A = {1, 2, 3, 4}, B = {3, 4, 5}, then:
A ∪ B = {1, 2, 3, 4, 5} → Students who play either
A ∩ B = {3, 4} → Students who play both
A relation R on a set is a subset of the Cartesian product of the set with itself. Common properties of relations:
Example:
A = {1, 2}
R = {(1,1), (2,2), (1,2), (2,1)}
- Reflexive: Yes
- Symmetric: Yes
- Transitive: Yes
An equivalence relation is a relation that is reflexive, symmetric, and transitive.
Example:
Let A = {1, 2, 3}
R = {(1,1), (2,2), (3,3), (1,2), (2,1)}
Not transitive → Not an equivalence relation
A partial order relation is reflexive, antisymmetric, and transitive.
Example:
A = {1, 2, 3}, R = {(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)}
This is a partial order relation.
A function f from A to B assigns exactly one element of B to each element of A.
Example:
f(x) = x²
Domain = {1, 2, 3}, Range = {1, 4, 9}
Example:
f: A → B, A = {1, 2}, B = {a, b, c}
f(1)=a, f(2)=b → This is "into" (c is unused)
Given f: A → B and g: B → C, the composite function g∘f is defined as: (g∘f)(x) = g(f(x))
Example:
f(x) = x + 2, g(x) = 3x
(g∘f)(x) = g(x + 2) = 3(x + 2) = 3x + 6
An inverse function f−1 reverses the mapping of f. f(f−1(x)) = x
Example:
f(x) = 2x + 3
Then, f⁻¹(x) = (x - 3)/2
Functions like sin(x), cos(x), tan(x) that relate angles to ratios of sides in right triangles.
Example:
If angle = 30°
sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3
The inverse of the exponential function. If ax = y, then loga(y) = x
Example:
log₂(8) = 3, because 2³ = 8
An exponential function has the form f(x) = ax, where a is a constant.
Example:
f(x) = 2^x
f(1) = 2, f(2) = 4, f(3) = 8
A partial order set (poset) is a set combined with a relation that is reflexive, antisymmetric, and transitive.
Example:
Set A = {1, 2, 3}
Relation R = {(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)}
R is reflexive, antisymmetric, and transitive → So (A, R) is a poset.
A Hasse diagram is a graphical way to represent a poset by removing reflexive and implied edges, and drawing elements in a way that lower elements are placed below higher ones.
Example:
Set A = {1, 2, 4, 8} with relation "divides"
Hasse diagram:
1
|
2
|
4
|
8
A chain in a poset is a subset in which every pair of elements is comparable (i.e., for any a and b, either a ≤ b or b ≤ a).
Example:
In set A = {1, 2, 3, 4} with usual ≤ relation,
{2, 3, 4} is a chain since all elements are comparable.
Minimal point: An element in a poset that has no other element less than it.
Maximal point: An element that has no other element greater than it.
Example:
Set A = {a, b, c} with R = {(a,a), (b,b), (c,c)}
Here, a, b, and c are both maximal and minimal because they are incomparable.
glb (also called meet): The largest element that is less than or equal to both.
lub (also called join): The smallest element that is greater than or equal to both.
Example:
Set A = {1, 2, 4, 8}, Relation: divides
glb(4, 8) = 4, lub(2, 4) = 4
A lattice is a poset where every pair of elements has both a least upper bound (lub) and a greatest lower bound (glb).
It is an algebraic system with two binary operations: meet ( ∧ ) and join ( ∨ ).
Example:
Set A = {1, 2, 4, 8} with divides relation
- glb(2, 4) = 2, lub(2, 4) = 4 → So it's a lattice
The Principle of Duality states that every statement in lattice theory remains true if we interchange ∧ (meet) with ∨ (join), and ≤ with ≥.
Example:
Original: a ≤ b implies a ∧ b = a
Dual: a ≥ b implies a ∨ b = a
Both are valid due to duality.
A determinant is a scalar value that can be computed from the elements of a square matrix and represents certain properties like volume scaling and invertibility.
Example:
|A| = | 1 2 |
| 3 4 | = (1×4) - (2×3) = 4 - 6 = -2
Minor: The minor of an element is the determinant formed by deleting its row and column.
Cofactor: Cofactor = (–1)i+j × Minor of element at (i, j).
Example:
Matrix: | 1 2 3 |
| 0 4 5 |
| 1 0 6 |
Minor of element (1,1): | 4 5 |
| 0 6 | = (4×6 - 5×0) = 24
Cofactor = (–1)1+1 × 24 = 24
A matrix is a rectangular array of numbers arranged in rows and columns.
Example:
A = |1 2|, B = |3 4| → A + B = |4 6|
Adjoint: Transpose of cofactor matrix.
Inverse: A−1 = adj(A) / det(A), if det(A) ≠ 0
Example:
A = |1 2|
|3 4|
Det(A) = -2
Adj(A) = | 4 -2 |
| -3 1 |
A−1 = (1/−2) × Adj(A)
Used to solve linear equations: A·X = B. Xi = det(Ai)/det(A)
Example:
2x + 3y = 5
x + 2y = 4
A = |2 3|
|1 2| → det = 1
Replace first column with B: |5 3| → det = 4 → x = 4/1 = 4
|4 2|
Replace second column with B: |2 5| → det = 3 → y = 3/1 = 3
|1 4|
The rank is the maximum number of linearly independent rows or columns.
It helps in solving systems of linear equations and determining consistency.
Vectors are linearly dependent if one can be written as a combination of others.
If no such combination exists, they are independent.
If A·v = λ·v, then λ is the eigenvalue and v is the eigenvector.
They describe directions that remain unchanged under linear transformation.
This theorem states that every square matrix satisfies its own characteristic equation.
Note: This is stated without proof.
Example:
A = |2 1|
|1 2|
Characteristic equation: λ² - 4λ + 3 = 0
Cayley-Hamilton says: A² - 4A + 3I = 0
The limit of a function f(x) as x approaches a value a is the value the function approaches as the input approaches a.
Written as: limx→a f(x) = L
Example:
limx→2 (3x + 1) = 3(2) + 1 = 7
Limits can be evaluated for:
limx→0 (sin x)/x = 1
limx→0 (ex - 1)/x = 1
Example:
limx→0 (sin x)/x = 1
A function f(x) is continuous at a point x = a if:
Example:
f(x) = x² is continuous at x = 2
→ limx→2 f(x) = f(2) = 4
A function is continuous over [a, b] if it is continuous at every point in the interval and both one-sided limits at a and b exist and match function values:
If a function f is continuous on a closed interval [a, b] and k is any value between f(a) and f(b), then there exists at least one c ∈ [a, b] such that:
f(c) = k
Example:
f(x) = x³, f(1) = 1, f(2) = 8
Then f(x) takes value 5 for some x ∈ [1, 2]
Example of jump:
f(x) = {
1, x < 0
2, x ≥ 0
}
→ Discontinuous at x = 0 (jump)
The derivative of a function shows how the function changes as the input changes. It is the slope of the tangent line to the curve at a point.
It is written as: f′(x) or dy/dx
Example:
If f(x) = x², then f′(x) = 2x
Example:
If f(x) = x² and g(x) = x,
Then (f × g)′ = (x² × x)′ = 2x × x + x² × 1 = 3x²
The chain rule is used when one function is inside another (composite functions).
If y = f(g(x)), then: dy/dx = f′(g(x)) × g′(x)
Example:
If y = (3x + 1)², then dy/dx = 2(3x + 1) × 3 = 6(3x + 1)
Composite functions involve functions inside other functions. Use the chain rule to find their derivatives.
Example:
f(x) = sin(x²), then f′(x) = cos(x²) × 2x
Used when differentiating complicated products, quotients, or powers. Take the natural log of both sides first.
Example:
y = x^x
ln y = ln(x^x) = x ln x
Then, d(ln y)/dx = ln x + 1 ⇒ dy/dx = x^x (ln x + 1)
The definite integral can be seen as the limit of a sum of areas of rectangles under a curve as the number of rectangles approaches infinity.
Written as: ∫ab f(x) dx = limn→∞ Σ f(xi) Δx
The theorem connects differentiation and integration. It states that if F is the antiderivative of f, then:
∫ab f(x) dx = F(b) - F(a)
Indefinite integral is the reverse of derivative. It gives a family of functions that differ by a constant.
Written as: ∫ f(x) dx = F(x) + C, where C is the constant of integration.
Example:
∫ x² dx = (1/3)x³ + C
Formula: ∫ u dv = uv - ∫ v du
Example (Substitution):
∫ 2x(x² + 1) dx → Let u = x² + 1, then du = 2x dx
→ ∫ u du = (1/2)u² + C = (1/2)(x² + 1)² + C
Example (By Parts):
∫ x eˣ dx → u = x, dv = eˣ dx
→ ∫ x eˣ dx = x eˣ - ∫ eˣ dx = x eˣ - eˣ + C