A set is a well-defined collection of distinct objects, considered as an object in its own right. The objects in a set are called its elements or members. Sets are usually represented using curly braces, e.g., {a, b, c}.
Example:
Set A = {1, 2, 3, 4, 5}
Set B = {x | x is a vowel in the English alphabet} = {a, e, i, o, u}
A set A is a subset of set B if all elements of A are also elements of B. This is denoted as A ⊆ B.
Example:
A = {1, 2, 3}, B = {1, 2, 3, 4, 5}
Since all elements of A are in B, A ⊆ B.
Two sets are equal if they have the same elements, irrespective of the order. This is denoted as A = B.
Example:
A = {1, 2, 3}, B = {3, 2, 1}
Since both sets have the same elements, A = B.
The universal set is the set that contains all the elements under consideration, typically denoted by U. All other sets are subsets of the universal set.
Example:
U = {1, 2, 3, 4, 5}, A = {1, 2}, B = {4, 5}
Here, U is the universal set.
A set is finite if it has a limited number of elements. Otherwise, it is called infinite.
Example:
Finite Set: A = {1, 2, 3}
Infinite Set: B = {x | x is a natural number}
Operations such as Union, Intersection, and Complement are commonly performed on sets:
Example:
A = {1, 2, 3}, B = {3, 4, 5}, U = {1, 2, 3, 4, 5}
A ∪ B = {1, 2, 3, 4, 5}
A ∩ B = {3}
A' = {4, 5}
The Cartesian product of two sets A and B, denoted as 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 refers to the number of elements in the set, denoted as |A|.
Example:
A = {1, 2, 3, 4}
|A| = 4
Sets are widely used in various fields such as computer science, mathematics, and logic. They are used for database operations, defining groups in algebra, and problem-solving in probability and statistics.
Example:
Problem: Find students enrolled in both Math and Science.
Math = {John, Alice, Bob}
Science = {Alice, Bob, Clara}
Math ∩ Science = {Alice, Bob}
A relation is a subset of the Cartesian product of two sets, describing how elements of the first set are related to elements of the second set. Relations have various properties, such as:
a in set A, (a, a) ∈ R.(a, b) ∈ R, then (b, a) ∈ R.(a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
Example:
A = {1, 2, 3}, R = {(1, 1), (2, 2), (2, 3), (3, 2)}
Reflexive: (1, 1), (2, 2)
Symmetric: (2, 3) and (3, 2)
Transitive: No violation observed.
A relation is an equivalence relation if it is reflexive, symmetric, and transitive. Equivalence relations partition a set into equivalence classes.
Example:
A = {1, 2, 3}, R = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}
R is reflexive, symmetric, and transitive. Hence, R is an equivalence relation.
A relation is a partial order relation if it is reflexive, antisymmetric, and transitive. It defines a partial ordering among elements.
Example:
A = {1, 2, 3}, R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3)}
R is reflexive, antisymmetric, and transitive. Hence, R is a partial order relation.
A function is a special relation where each element of the domain is associated with exactly one element of the codomain. The domain is the set of all input values, and the range is the set of all output values.
Example:
f(x) = x²
Domain: All real numbers
Range: Non-negative real numbers
Functions can be classified based on their mapping:
Example:
f(x) = x²
Domain: {1, 2, 3}
Codomain: {1, 4, 9, 16}
Range: {1, 4, 9}
This is an "into" function.
The composite function of two functions f and g is defined as (f∘g)(x) = f(g(x)). The inverse function of f, denoted as f⁻¹, reverses the mapping of f.
Example:
f(x) = 2x, g(x) = x + 1
Composite: (f∘g)(x) = f(g(x)) = 2(x + 1) = 2x + 2
Inverse: If f(x) = 2x, then f⁻¹(x) = x / 2
Trigonometric functions are based on angles and include sine, cosine, tangent, and their reciprocals. They are widely used in geometry and periodic functions.
Example:
sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3
Logarithmic functions are the inverses of exponential functions. The general form is log_b(x), where b is the base. Exponential functions are of the form a^x.
Example:
Exponential: f(x) = 2^x, f(3) = 2³ = 8
Logarithmic: g(x) = log₂(x), g(8) = 3
A partial order set (poset) is a set paired with a relation that is reflexive, antisymmetric, and transitive. A poset provides a partial ordering of elements.
Example:
Set A = {1, 2, 3}
Relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3)}
R is reflexive, antisymmetric, and transitive, so (A, R) is a poset.
A Hasse diagram is a graphical representation of a poset. It shows the ordering of elements with minimal edges while omitting self-loops and transitive edges.
Example:
Set A = {a, b, c}, Relation R = {(a, a), (b, b), (c, c), (a, b), (a, c)}
Hasse Diagram:
a
↑
b c
A chain is a subset of a poset where every pair of elements is comparable under the given relation.
Example:
Poset: A = {1, 2, 3}, R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}
Chain: {1, 2, 3} (as 1 < 2 < 3)
A point in a poset is maximal if no other element is greater than it, and minimal if no other element is smaller than it.
Example:
A = {a, b, c}, R = {(a, b), (a, c)}
Minimal Point: a
Maximal Points: b, c
The greatest lower bound (glb) of a subset is the largest element less than or equal to all elements in the subset. The least upper bound (lub) is the smallest element greater than or equal to all elements in the subset.
Example:
Set A = {a, b, c}, R = {(a, b), (a, c), (b, c)}
glb(b, c) = b
lub(a, b) = b
A lattice is a poset in which every pair of elements has both a lub and a glb. Lattices are algebraic structures that satisfy the lattice axioms.
Example:
A = {a, b, c}, lub(a, b) = b, glb(a, b) = a
Hence, (A, R) is a lattice.
The principle of duality in lattice theory states that every lattice property has a dual property, which can be obtained by interchanging "join" and "meet" operations.
Example:
Dual of (A ∪ B) = A ∩ B
A sublattice is a subset of a lattice that is itself a lattice under the same join and meet operations as the original lattice.
Example:
Original Lattice: A = {a, b, c, d}, Join: ∪, Meet: ∩
Sublattice: {a, b, c}
A lattice is distributive if the meet and join operations distribute over each other. A lattice is complemented if every element has a complement that satisfies specific conditions.
Example:
Distributive Lattice: A = {0, 1, a, b}, (a ∩ (b ∪ c)) = (a ∩ b) ∪ (a ∩ c)
Complemented Lattice: A = {0, 1}, Complement of 0 is 1 and vice versa.
Partial differentiation is the process of finding the derivative of a function with respect to one variable while keeping the other variables constant. It is used to study functions of multiple variables.
Example:
f(x, y) = x²y + y³
Partial derivative with respect to x:
∂f/∂x = 2xy
Partial derivative with respect to y:
∂f/∂y = x² + 3y²
Change of variables involves substituting one set of variables with another to simplify a given function or solve an equation.
Example:
Given: u = x + y, v = x - y
Function: f(x, y) = x² - y²
Substitution: f(u, v) = (u + v)²/4 - (u - v)²/4
Simplified: f(u, v) = uv
The chain rule is a formula to compute the derivative of a composite function. For functions of multiple variables, it accounts for indirect dependencies.
Example:
z = f(x, y), x = g(t), y = h(t)
Chain rule: dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)
If f(x, y) = x²y, x = t², y = t + 1:
dz/dt = 2xt² + t²
Substitute: dz/dt = 2(t²)t + t² = 2t³ + t²
To find the extrema (maximum or minimum values) of a function of two variables, we use partial derivatives and solve the critical point conditions:
∂f/∂x = 0 and ∂f/∂y = 0.
Example:
f(x, y) = x² + y² - 4x - 6y + 13
∂f/∂x = 2x - 4, ∂f/∂y = 2y - 6
Critical point: (2, 3)
Second derivatives: fxx = 2, fyy = 2, fxy = 0
H = (2)(2) - 0² = 4 > 0 and fxx > 0
Minimum point at (2, 3)
Euler’s theorem states that for a homogeneous function of degree n, the following relation holds:
f(x, y, ...) = x(∂f/∂x) + y(∂f/∂y) + ...
A function f(x, y) is homogeneous of degree n if:
f(tx, ty) = tⁿf(x, y)
Example:
f(x, y) = x²y (homogeneous of degree 3)
x(∂f/∂x) + y(∂f/∂y) = x(2xy) + y(x²) = 3x²y
Satisfies Euler’s theorem.
In 3D Coordinate Geometry, a point is represented by its coordinates (x, y, z) in a 3D space, where x, y, and z are distances along the respective axes.
Example:
A point P(2, 3, 5) is located at 2 units along x-axis, 3 units along y-axis, and 5 units along z-axis.
The direction cosines of a line are the cosines of the angles it makes with the coordinate axes. The angle between two lines can be found using:
cos(θ) = (l₁l₂ + m₁m₂ + n₁n₂) / √(l₁² + m₁² + n₁²) √(l₂² + m₂² + n₂²)
where l, m, n are direction cosines.
Example:
For lines L₁ (2, 3, -1) and L₂ (1, -1, 2):
cos(θ) = (2×1 + 3×(-1) + (-1)×2) / √(2² + 3² + (-1)²)√(1² + (-1)² + 2²)
Result: cos(θ) = -0.1
The projection of the line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) onto a plane can be calculated using vector geometry.
Example:
P(1, 2, 3), Q(4, 6, 9)
Direction vector: PQ = (3, 4, 6)
Projection onto xy-plane = √(3² + 4²) = 5
The equation of a plane can be written as ax + by + cz + d = 0, where (a, b, c) is the normal vector to the plane.
Example:
Plane passing through (1, 2, 3) with normal vector (2, -1, 4):
2(x - 1) - 1(y - 2) + 4(z - 3) = 0
Simplified: 2x - y + 4z - 14 = 0
The parametric equation of a straight line in 3D space is:
x = x₁ + λa, y = y₁ + λb, z = z₁ + λc
where (a, b, c) is the direction vector and (x₁, y₁, z₁) is a point on the line.
Example:
Line through (1, 2, 3) with direction vector (2, 1, -1):
x = 1 + 2λ, y = 2 + λ, z = 3 - λ
A line lies on a plane if its direction vector satisfies the plane equation, and the given point of the line also satisfies the plane equation.
Example:
Line: x = 1 + 2λ, y = 2 + λ, z = 3 - λ
Plane: 2x - y + z = 7
Check: Substituting x, y, z:
2(1 + 2λ) - (2 + λ) + (3 - λ) = 7
Simplified: λ satisfies the equation, so the line lies on the plane.
Two lines are coplanar if the scalar triple product of their direction vectors and the vector connecting points on the two lines is zero.
Example:
Lines: L₁ and L₂ with direction vectors d₁ and d₂, and points P₁, P₂:
Scalar triple product: (P₂ - P₁) • (d₁ × d₂) = 0
If true, the lines are coplanar.
The shortest distance between two skew lines is given by:
D = |(P₂ - P₁) • (d₁ × d₂)| / |d₁ × d₂|
Example:
Lines: L₁: (1, 2, 3) + λ(2, 1, -1), L₂: (4, 6, 9) + μ(1, -1, 2)
Shortest distance D = ...
The equation of a sphere is (x - x₀)² + (y - y₀)² + (z - z₀)² = r², where (x₀, y₀, z₀) is the center and r is the radius.
Example:
Sphere with center (2, 3, 5) and radius 4:
(x - 2)² + (y - 3)² + (z - 5)² = 16
The equation of the tangent plane to a sphere at a point (x₁, y₁, z₁) is:
(x - x₀)(x₁ - x₀) + (y - y₀)(y₁ - y₀) + (z - z₀)(z₁ - z₀) = 0
Example:
Sphere: (x - 2)² + (y - 3)² + (z - 5)² = 16
Point: (3, 5, 7)
Tangent plane: (x - 2)(3 - 2) + (y - 3)(5 - 3) + (z - 5)(7 - 5) = 0
Simplified: x + 2y + 2z = 22
A double integral in Cartesian coordinates is used to calculate the area under a surface in a given region. It is represented as:
∫∫_R f(x, y) dx dy
where R is the region of integration.
Example:
Evaluate ∫∫_R (x + y) dx dy, where R is bounded by x = 0, x = 2, y = 0, y = 1.
Solution:
∫ (from x=0 to 2) ∫ (from y=0 to 1) (x + y) dy dx
= ∫ (from x=0 to 2) [xy + y²/2] (from y=0 to 1) dx
= ∫ (from x=0 to 2) [x(1) + (1)²/2 - 0] dx
= ∫ (from x=0 to 2) (x + 0.5) dx
= [x²/2 + 0.5x] (from x=0 to 2)
= 4.5
In polar coordinates, a double integral is expressed as:
∫∫_R f(r, θ) r dr dθ
where r is the radius and θ is the angle.
Example:
Evaluate ∫∫_R r² dr dθ, where R is the region bounded by r=0 to 1 and θ=0 to π/2.
Solution:
∫ (from θ=0 to π/2) ∫ (from r=0 to 1) r² dr dθ
= ∫ (from θ=0 to π/2) [r³/3] (from r=0 to 1) dθ
= ∫ (from θ=0 to π/2) (1/3) dθ
= (θ/3) (from θ=0 to π/2)
= π/6
The order of integration in a double integral can be changed to simplify the calculation, provided the integration limits are adjusted correctly.
Example:
Evaluate ∫ (from x=0 to 1) ∫ (from y=x to 1) f(x, y) dy dx.
Change the order of integration:
∫ (from y=0 to 1) ∫ (from x=0 to y) f(x, y) dx dy.
A triple integral in Cartesian coordinates is used to calculate the volume of a solid region. It is represented as:
∫∫∫_V f(x, y, z) dx dy dz
where V is the volume of integration.
Example:
Evaluate ∫∫∫_V 1 dx dy dz, where V is bounded by x=0 to 1, y=0 to x, z=0 to y.
Solution:
∫ (from x=0 to 1) ∫ (from y=0 to x) ∫ (from z=0 to y) 1 dz dy dx
= ∫ (from x=0 to 1) ∫ (from y=0 to x) [z] (from z=0 to y) dy dx
= ∫ (from x=0 to 1) ∫ (from y=0 to x) y dy dx
= ∫ (from x=0 to 1) [y²/2] (from y=0 to x) dx
= ∫ (from x=0 to 1) (x²/2) dx
= [x³/6] (from x=0 to 1)
= 1/6
Triple integrals can be used to find the volume of simple shapes, such as spheres, cylinders, and cubes.
Example:
Find the volume of a sphere of radius R.
Volume = ∫∫∫_V 1 dx dy dz
In spherical coordinates: ∫ (from θ=0 to 2π) ∫ (from φ=0 to π) ∫ (from r=0 to R) r² sin(φ) dr dφ dθ
= 4πR³/3