Class 12 Maths proofs: the types to know

An equivalence relation

Show a relation is reflexive, symmetric and transitive · Chapter 1: Relations and Functions

1. Reflexive: for every integer \(a\), \(a - a = 0 = 4 \times 0\), so \((a, a) \in R\).
2. Symmetric: if \((a, b) \in R\) then \(a - b = 4k\) for some integer \(k\), so \(b - a = 4(-k)\) and \((b, a) \in R\).
3. Transitive: if \(a - b = 4k\) and \(b - c = 4l\), then \(a - c = 4(k + l)\), so \((a, c) \in R\).
4. \(R\) is reflexive, symmetric and transitive, so it is an equivalence relation.

Where marks slip: Each property is a separate claim: write the general element (\(a\), \(b\), \(c\) any integers), not one example.

One-one and onto

Show a function is a bijection (and spot one that is not) · Chapter 1: Relations and Functions

1. One-one: if \(f(a) = f(b)\) then \(5a - 3 = 5b - 3\), so \(a = b\).
2. Onto: for any real \(y\), take \(x = \dfrac{y + 3}{5}\), which is real; then \(f(x) = 5 \cdot \dfrac{y + 3}{5} - 3 = y\).
3. For \(g\): \(g(-1) = g(1) = 1\) with \(-1 \neq 1\), so \(g\) is not one-one.
4. \(g(x) = x^2 \geq 0\) for every real \(x\), so \(-1\) has no pre-image and \(g\) is not onto.

Where marks slip: To show something fails, one counter-example is enough. To show it holds, argue for every element.

The inverse of a matrix is unique

Proof from the definition · Chapter 3: Matrices

1. Suppose \(B\) and \(C\) are both inverses of \(A\): \(AB = BA = I\) and \(AC = CA = I\).
2. Then \(B = BI = B(AC)\).
3. Matrix multiplication is associative, so \(B(AC) = (BA)C = IC = C\).
4. So \(B = C\): the inverse, when it exists, is unique.

Symmetric and skew-symmetric parts

Show a matrix property using transposes · Chapter 3: Matrices

1. \((A + A^{T})^{T} = A^{T} + (A^{T})^{T}\) \(= A^{T} + A = A + A^{T}\), so \(A + A^{T}\) is symmetric.
2. \[\begin{aligned}(A - A^{T})^{T} &= A^{T} - A \\ &= -(A - A^{T})\end{aligned}\], so \(A - A^{T}\) is skew-symmetric.

Where marks slip: Quote the rules you use: \((P + Q)^{T} = P^{T} + Q^{T}\) and \((P^{T})^{T} = P\).

Continuous but not differentiable

Show continuity and differentiability at a point · Chapter 5: Continuity and Differentiability

1. Continuity: \(\lim_{x \to 3^-} |x - 3| = 0\), \(\lim_{x \to 3^+} |x - 3| = 0\) and \(f(3) = 0\). All three are equal, so \(f\) is continuous at \(3\).
2. Left-hand derivative: \[\lim_{h \to 0^-} \dfrac{f(3 + h) - f(3)}{h}\] \(= \lim_{h \to 0^-} \dfrac{|h|}{h} = -1\).
3. Right-hand derivative: \(\lim_{h \to 0^+} \dfrac{|h|}{h} = 1\).
4. \(-1 \neq 1\), so \(f\) is not differentiable at \(x = 3\).

Where marks slip: Write all three values for continuity (left limit, right limit, \(f(3)\)), and both one-sided derivatives.

A strictly increasing function

Show a function is increasing using the derivative · Chapter 6: Application of Derivatives

1. \(f'(x) = 3x^2 + 2\).
2. \(x^2 \geq 0\) for every real \(x\), so \(f'(x) \geq 2 > 0\) for every real \(x\).
3. So \(f\) is strictly increasing on \(\mathbb{R}\).

Verifying a solution of a differential equation

Show a function satisfies a differential equation · Chapter 9: Differential Equations

1. \(\dfrac{dy}{dx} = 2Ae^{2x} - 3Be^{-3x}\) and \[\dfrac{d^2y}{dx^2} = 4Ae^{2x} + 9Be^{-3x}\]
2. LHS \(= (4Ae^{2x} + 9Be^{-3x})\) \(+ (2Ae^{2x} - 3Be^{-3x})\) \(- 6(Ae^{2x} + Be^{-3x})\)
3. \(= (4 + 2 - 6)Ae^{2x}\) \(+ (9 - 3 - 6)Be^{-3x}\) \(= 0 =\) RHS.

Three collinear points

Show points are collinear with vectors · Chapter 10: Vector Algebra

1. \[\overrightarrow{AB} = \hat{i} + 2\hat{j} + 2\hat{k}\] and \[\overrightarrow{AC} = 3\hat{i} + 6\hat{j} + 6\hat{k}\]
2. \[\overrightarrow{AC} = 3\,\overrightarrow{AB}\], so the two vectors are parallel.
3. They share the point \(A\), so \(A\), \(B\) and \(C\) lie on one line.

Where marks slip: Parallel alone is not enough: say that the vectors have a common point.