Important Proofs & Theorems

Given an arbitrary equivalence relation R ∈ an arbitrary set X, R divides X into mutually disjoint subsets Aᵢ called partitions or subdivisions of X satisfying: (i) all elements of Aᵢ are related to each other, for all i, (ii) no element of Aᵢ is related to any element of Aⱼ, i ≠ j, (iii) union of Aⱼ = X and Aᵢ intersect Aⱼ = φ, i ≠ j. The subsets Aᵢ are called equivalence classes.

For an arbitrary finite set X, a one-one function f: X → X is necessarily onto and an onto map f: X → X is necessarily one-one. This is a characteristic difference between a finite and an infinite set.

A function f is invertible if and only if f is one-one and onto (bijective). If f is invertible, then f must be one-one and onto, and conversely, if f is one-one and onto, then f must be invertible.

For any square matrix A with real number entries, A + A' is a symmetric matrix and A - A' is a skew symmetric matrix.

Any square matrix can be expressed as the ∑ of a symmetric and a skew symmetric matrix. Specifically, A = (1/2)(A + A') + (1/2)(A - A').

Inverse of a square matrix, if it exists, is unique.

If A and B are invertible matrices of the same order, then (AB)⁻¹ = B⁻¹ A⁻¹.

If A be any given square matrix of order n, then A(adj A) = (adj A)A = |A|*I, where I is the identity matrix of order n.

If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.

The determinant of the ∏ of matrices is equal to ∏ of their respective determinants, that is, |AB| = |A| · |B|, where A and B are square matrices of the same order.

A square matrix A is invertible if and only if A is nonsingular matrix.

Suppose f and g be two real functions continuous at a real number c. Then: (1) f + g is continuous at x = c, (2) f - g is continuous at x = c, (3) f . g is continuous at x = c, (4) f/g is continuous at x = c (provided g(c) ≠ 0)

Suppose f and g are real valued functions such that (f o g) is defined at c. If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c

If a function f is differentiable at a point c, then it is also continuous at that point.

Every differentiable function is continuous. The converse is NOT true: f(x) = |x| is continuous at x = 0 but not differentiable at x = 0.

Let f be a real valued function which is a composite of two functions u and v; i.e., f = v o u. Suppose t = u(x) and if both dt/dx and dv/dt exist, then df/dx = (dv/dt) . (dt/dx)

(1) The derivative of eˣ w.r.t. x is eˣ; i.e., d/dx(eˣ) = eˣ. (2) The derivative of log x w.r.t. x is 1/x; i.e., d/dx(log x) = 1/x. (*Please see supplementary material on Page 222)

If f: [a, b] → R is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists some c ∈ (a, b) such that f'(c) = 0

If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b), then there exists some c ∈ (a, b) such that f'(c) = [f(b) - f(a)]/(b - a)

Let f be continuous on [a, b] and differentiable on (a, b). Then: (a) f is increasing ∈ [a,b] if f'(x) ≥ 0 for each x ∈ (a,b) ┃ (b) f is decreasing ∈ [a,b] if f'(x) ≤ 0 for each x ∈ (a,b) ┃ (c) f is a constant function ∈ [a,b] if f'(x) = 0 for each x ∈ (a,b).

Let f be a function defined on an open interval I. Suppose c is ∈ I. If f has a local maxima or a local minima at x = c, then either f'(c) = 0 or f is not differentiable at c.

Let f be a function defined on an open interval I. Let f be continuous at a critical point c ∈ I. Then: (i) If f'(x) changes sign from positive to negative as x increases through c (f'(x) > 0 to left, f'(x) < 0 to right), then c is a point of local maxima. (ii) If f'(x) changes sign from negative to positive as x increases through c (f'(x) < 0 to left, f'(x) > 0 to right), then c is a point of local minima. (iii) If f'(x) does not change sign as x increases through c, then c is neither a point of local maxima nor local minima (point of inflection).

Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then: (i) x = c is a point of local maxima if f'(c) = 0 and f''(c) < 0. The value f(c) is local maximum value. (ii) x = c is a point of local minima if f'(c) = 0 and f''(c) > 0. The value f(c) is local minimum value. (iii) The test fails if f'(c) = 0 and f''(c) = 0. In this case, go back to the first derivative test.

Let f be a continuous function on an interval I = [a, b]. Then f has the absolute maximum value and f attains it at least once ∈ I. Also, f has the absolute minimum value and attains it at least once ∈ I.

Let f be a differentiable function on a closed interval I and let c be any interior point of I. Then: (i) f'(c) = 0 if f attains its absolute maximum value at c. (ii) f'(c) = 0 if f attains its absolute minimum value at c.

Let f be a continuous function on the closed interval [a, b] and let A(x) be the area function. Then A'(x) = f(x), for all x ∈ [a, b].

Let f be continuous function defined on the closed interval [a, b] and F be an anti derivative of f. Then ∫ from a to b of f(x) dx = [F(x)] from a to b = F(b) - F(a).

For any two vectors a and b: a + b = b + a

For any three vectors a, b, c: (a + b) + c = a + (b + c)

For any two vectors a and b: |a . b| ≤ |a||b|

For any two vectors a and b: |a + b| ≤ |a| + |b|

If |a + b| = |a| + |b|, then |AC| = |AB| + |BC|, showing that A, B, C are collinear.

Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.

Let R be the feasible region for a linear programming problem, and let Z = ax + by be the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum value on R and each of these occurs at a corner point (vertex) of R.

For any two events E and F: P(E ∩ F) = P(E) . P(F|E) = P(F) . P(E|F), provided P(E) ≠ 0 and P(F) ≠ 0.

If E and F are independent events, then (a) E and F' are independent, (b) E' and F are independent, (c) E' and F' are independent.

Let {E₁, E₂, ..., Eₙ} be a partition of the sample space S, and suppose that each of E₁, E₂, ..., Eₙ has nonzero probability of occurrence. Let A be any event associated with S, then P(A) = P(E₁) P(A|E₁) + P(E₂) P(A|E₂) + ... + P(Eₙ) P(A|Eₙ) = sumⱼ=₁^{n} P(Eⱼ) P(A|Eⱼ).

If E₁, E₂, ..., Eₙ are n non-empty events which constitute a partition of sample space S, i.e., E₁, E₂, ..., Eₙ are pairwise disjoint and E₁ u E₂ u ... u Eₙ = S, and A is any event of nonzero probability, then P(Eᵢ|A) = P(Eᵢ) P(A|Eᵢ) / sumⱼ=₁^{n} P(Eⱼ) P(A|Eⱼ), for any i = 1, 2, 3, ..., n.