Key points

Reflexivity must hold for every element of the set — a single missing pair (a,a) destroys it.

The empty relation on a non-empty set is symmetric and transitive, but never reflexive.

The universal relation A \times A is always an equivalence relation.

To show a relation is transitive you must check every chain; to show it is not, one broken chain suffices.

Equivalence classes are either identical or completely disjoint — they never partially overlap.

Whether a function is onto depends on the codomain you were given, not on the formula alone.

A strictly increasing or strictly decreasing function is automatically one-one.

For a function from a finite set to itself, one-one and onto are equivalent. For infinite sets they are not.

A function is invertible exactly when it is bijective — this is what makes the next chapter possible.

\cos^{-1}, \cot^{-1} and \sec^{-1} never return a negative value — their ranges lie inside [0, \pi].

\sin^{-1}(\sin x) = x only when x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right].

\sin^{-1}x is not \dfrac{1}{\sin x} — the -1 is not an exponent.

\sin^{-1}x + \cos^{-1}x = \dfrac{\pi}{2} for every x \in [-1,1].

\tan^{-1}x approaches \pm\frac{\pi}{2} but never attains them — the range is an open interval.

\sin^{-1}x is undefined for |x| > 1. Watch for questions whose arguments quietly exceed 1.

y = \cos^{-1}x and y = \cot^{-1}x are decreasing functions; the other four inverse functions increase.

Matrix multiplication is not commutative: AB \neq BA in general, and often only one product is even defined.

AB = O does not imply A = O or B = O.

Cancellation is invalid: AB = AC does not give B = C unless A is invertible.

Every diagonal entry of a skew-symmetric matrix is zero.

(AB)^{T} = B^{T}A^{T} and (AB)^{-1} = B^{-1}A^{-1} — both reverse the order.

Addition requires the two matrices to have exactly the same order; multiplication does not.

The identity matrix I satisfies AI = IA = A — it is the one matrix that does commute with everything of its order.

Two matrices are equal only when they have the same order and every corresponding entry matches.

A^{-1} exists if and only if |A| \neq 0.

The adjoint is the transpose of the cofactor matrix, not the cofactor matrix itself.

Expand along whichever row or column has the most zeros — each zero kills an entire cofactor term.

|kA| = k^{n}|A| for an n \times n matrix, not k|A|.

Area must be positive — take the absolute value of the determinant before halving.

If |A| = 0, the system AX = B has either no solution or infinitely many — never exactly one.

Determinants multiply: |AB| = |A||B|, even though AB \neq BA in general.

Differentiability implies continuity, but continuity does not imply differentiability.

For a piecewise function, always compute LHL, RHL and f(a) separately and compare all three.

A function that is discontinuous at a point cannot be differentiable there — say so and stop.

In logarithmic differentiation, the left side becomes \dfrac{1}{y}\dfrac{dy}{dx}, not \dfrac{dy}{dx}.

Always substitute y back in terms of x at the end of logarithmic or implicit differentiation.

For a sum such as x^{x} + x^{\sin x}, differentiate each term separately — \ln(a+b) does not simplify.

For parametric second derivatives, remember the final factor \dfrac{1}{dx/dt}.

f'(c) = 0 makes c a candidate, not automatically an extremum — it may be a point of inflection.

For absolute extrema on a closed interval, always evaluate at the endpoints as well.

If f''(c) = 0, the second derivative test says nothing — switch to the first derivative test.

Reduce an optimisation problem to a single variable using the constraint before differentiating.

State the domain of the variable in an applied problem — lengths and radii are positive.

In related-rates questions, differentiate with respect to time and use the chain rule throughout.

'Strictly increasing' needs f'(x) > 0; f'(x) \geq 0 only gives 'increasing' (non-decreasing).

Never omit the constant of integration C in an indefinite integral.

\displaystyle\int \frac{1}{x}\,dx = \ln|x| + C — the modulus is part of the answer.

When substituting in a definite integral, change the limits to match the new variable.

Choose u for integration by parts using ILATE.

Check whether \int_{-a}^{a} has an odd integrand — if so the answer is 0 with no work at all.

Before using partial fractions, confirm the degree of the numerator is less than that of the denominator; otherwise divide first.

For \int \frac{dx}{ax^2+bx+c}, complete the square to reach one of the standard forms.

\displaystyle\int e^{x}[f(x) + f'(x)]\,dx = e^{x}f(x) + C — recognising this shape saves a full round of integration by parts.

Always sketch the region before writing any integral.

If the curve crosses the x-axis inside the interval, split the integral there and add the absolute values.

Area is always positive. A negative result means the region lies below the axis — take the modulus.

For area between curves, subtract lower from upper — and check which is which on the interval concerned.

Exploit symmetry for circles and ellipses: compute one quadrant and multiply.

Find the limits by solving the curves simultaneously, not by reading them off the question.

Classify before solving: separable, homogeneous, or linear. The form dictates the method.

Degree is undefined when a derivative sits inside a transcendental function or a radical.

In the substitution y = vx, the product rule gives \dfrac{dy}{dx} = v + x\dfrac{dv}{dx}.

The equation must be in the exact form \dfrac{dy}{dx} + Py = Q before you read off P.

Do not add a constant when computing the integrating factor.

A general solution of an nth-order equation contains exactly n arbitrary constants; a particular solution has none.

After solving a homogeneous equation, substitute v = \dfrac{y}{x} back — the answer must be in x and y.

The dot product gives a scalar; the cross product gives a vector. Check your answer is the right kind of object.

\vec{a}\cdot\vec{b} = 0 means perpendicular; \vec{a}\times\vec{b} = \vec{0} means parallel.

The cross product is anti-commutative: reversing the order reverses the direction.

The area of a triangle carries a factor \frac{1}{2}; the area of a parallelogram does not.

A projection is a signed scalar and may be negative — do not take its modulus unless asked.

Direction cosines always satisfy l^{2}+m^{2}+n^{2}=1; direction ratios need not.

Two lines in space may intersect, be parallel, or be skew — skew is the case with no plane containing both.

If the shortest distance between two lines comes out as zero, the lines intersect.

Check whether the direction vectors are parallel before choosing a distance formula.

Use the modulus in the angle formula — the angle between two lines is taken to be acute.

In the Cartesian form, the denominators are the direction ratios; the numerators must be x - x_1, not x + x_1.

Direction ratios are not unique — any non-zero multiple describes the same direction.

The optimum always occurs at a corner point of the feasible region.

For an unbounded region, a corner value is only the optimum if the extra half-plane test passes.

Never omit the non-negativity constraints x \geq 0, y \geq 0 from the formulation.

Find corner points by solving the boundary lines simultaneously — do not read them off the sketch.

If two adjacent corners give the same optimal value, every point on the segment between them is optimal.

Use the origin to test which side of a line to shade, unless the line passes through the origin.

P(A \mid B) and P(B \mid A) are different — the event already known goes after the bar.

Independent and mutually exclusive are not the same thing, and for events of non-zero probability they are incompatible.

In total probability and Bayes' theorem, the causes must be mutually exclusive and exhaustive.

The denominator in Bayes' theorem is the total probability of the observed event.

Name the events in words before writing any numbers — most errors in this chapter are set-up errors.

For 'at least one', it is nearly always faster to compute the complement.