May 2026 Mathematics I — Weeks 1–4 study guide

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2026-07-18T00:00:00.000Z

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A structured foundation for sets, functions, algebraic reasoning, and early calculus ideas. # Mathematics I, Weeks 1–4: definitions drive the method Mathematics rewards precision.

Mathematics I, Weeks 1–4: definitions drive the method

Mathematics rewards precision. Begin with the object’s definition, its allowed inputs, and the claim to be shown. Symbols compress meaning; they do not replace it.

Week 1 — sets, logic, and statements

A set is a collection of distinct objects. Membership (x ∈ A), subset (A ⊆ B), union, intersection, complement, and Cartesian product describe relationships between collections. A proposition is a statement that is true or false; a predicate becomes a proposition once its variable is specified. Quantifiers matter: “for every” and “there exists” reverse how counterexamples work.

To disprove a universal statement, one valid counterexample is enough. To prove it, the reasoning must cover every permitted input.

Week 2 — functions and domains

A function maps each input in its domain to exactly one output. Domain, codomain, and range are different: the range is the outputs actually reached. A relation fails to be a function when one allowed input has multiple outputs. Composition applies one function’s output as another’s input, so its domain must respect both rules.

Retrieval prompt: write a function with a denominator. State the excluded input before simplifying any expression.

Week 3 — algebra as transformation

An equation asserts equality; solving applies operations that preserve its solution set. Factorisation, expansion, completing a square, and exponent laws are tools, not goals. At each line ask whether the transformation is reversible; squaring, multiplying by an expression that could be zero, or cancelling a variable can introduce or lose solutions.

Test each candidate solution in the original statement, especially after division or squaring.

Week 4 — rates of change and calculus foundations

A limit describes values approached near an input; it is not necessarily the function’s value at that input. The derivative is an instantaneous rate of change, formalised as a limit of average rates. Graphically it is tangent slope; dimensionally it is output-units per input-unit. Differentiation rules are useful only after the function, domain, and interpretation are clear.

Retrieval prompt: give one real quantity and identify its derivative’s units. Explain why a derivative can be zero without the original function being zero.

The mathematical vocabulary map

Term	Definition	Useful distinction
statement / proposition	A sentence with a definite truth value.	A question or command is not a proposition.
predicate	A statement containing a variable, becoming true or false when the variable is specified.	`x > 3` is a predicate; `5 > 3` is a proposition.
implication	A conditional claim, “if P then Q”.	Its converse is `Q → P`; its contrapositive is `not Q → not P`.
domain	Inputs for which a function is defined.	The domain is not automatically every real number.
codomain	Declared target set of a function.	The range is the subset actually reached.
equivalent transformation	A step preserving exactly the same solution set.	Squaring or multiplying by a possibly-zero expression may fail to be equivalent.
limit	A value approached by a function near an input.	It can exist even if the function is undefined there.
derivative	The limiting rate of change of a function.	It is a new function or value, not merely “a slope formula”.

Proof and counterexample: the first discipline

Quantifiers decide the method. To prove a claim beginning for every, your reasoning must work for an arbitrary permitted input. To disprove it, one valid counterexample is enough. To prove a claim beginning there exists, construct one explicit example.

For an implication P → Q, avoid proving the converse by accident. A clean strategy is often the contrapositive: assume not Q, then logically derive not P. The original implication and its contrapositive always have the same truth value; the converse need not.

Original worked example: domain before simplification

Consider


text
f(x) = (x² - 9) / (x - 3)

Factoring gives (x - 3)(x + 3)/(x - 3), which simplifies to x + 3 only where x ≠ 3. The original expression is not defined at 3, so its domain is all real numbers except 3. The simplified rule describes the same outputs on the allowed inputs, but it does not repair the original hole. This is why the domain belongs at the top of the solution, not as an afterthought.

Function taxonomy — know which question you are answering

Injective (one-to-one): different inputs never share an output. A horizontal-line test is a graphical clue, not a substitute for reasoning.
Surjective (onto): every declared codomain value is attained by at least one domain input.
Bijective: both injective and surjective; therefore an inverse function exists on the stated sets.
Composite function: f(g(x)); first confirm that outputs of g lie in the domain of f.
Inverse relation: reverses input/output pairs. It becomes an inverse function only when the original is one-to-one on the given domain.

When a question asks for a range, do not merely swap symbols. Start with the domain, apply the transformation, and ask which output values are impossible.

Algebra as a chain of permissions

Every line in algebra should answer: what operation was applied, and is it reversible under the current conditions?

For example, from x² = 9 we obtain x = 3 or x = -3; taking a square root yields two possibilities. From x(x - 2) = 0, the zero-product property gives two branches. By contrast, dividing both sides by x is unsafe until you separately handle x = 0.

Original worked example: a rate has units

If s(t) is distance in kilometres and t is time in hours, then s'(t) has units kilometres per hour. If s'(4) = 0, the distance function has a horizontal tangent at hour 4; it does not say the distance travelled is zero. Units keep the calculus interpretation attached to the symbols.

Limit and derivative checklist

State the input value being approached.
Ask what happens from the left and the right when relevant.
Distinguish the limiting value from the function value at that point.
For a derivative, identify the changing quantity, the input quantity, and the resulting units.
Check whether the domain permits the input before reporting a numerical answer.

Four habits of mathematical precision

Track the domain first. An algebraic rewrite can preserve an expression only on values where both sides are defined. Cancelling x - 2 from (x² - 4)/(x - 2) gives x + 2 for x ≠ 2; it does not silently make the original expression defined at 2.
Distinguish implication from equivalence. If P → Q, every case satisfying P satisfies Q; the reverse may fail. “A number is divisible by 4” implies “it is even,” but an even number need not be divisible by 4.
Name the quantifier. “For every” requires a proof or a general argument; “there exists” is established by one valid witness. A counterexample refutes a universal statement, not an existential one.
Keep units through a rate. If distance is in kilometres and time in hours, the derivative has kilometres per hour. A derivative without units can be algebraically right yet physically misinterpreted.

Counterexample workshop

The statement “if ab = 0, then a = 0 and b = 0” is false. Take a = 0, b = 5. The correct conclusion is that at least one factor is zero.
The graph of y = x² has a derivative of zero at x = 0, but that does not mean the function is constant near zero. A zero instantaneous rate is local information, not a global description.
A relation can pair values without being a function. If one input is assigned two outputs, it fails the one-output rule even if each individual pair looks reasonable.

Original retrieval lab

 Explain-back checkpoint
Can you make a derivative as a local rate useful without copying the chapter?
Write from memory first. This saves only on this device and never declares an answer correct; it makes the gap visible so you know exactly what to reread.
Explain the difference between an average rate and a derivative, then name one situation where the derivative alone does not describe the whole function.
Your recall draft
0/3 parts drafted
MeaningDefine it in ordinary language.MechanismWhat causes it or how does it work?BoundaryGive a caveat, non-example, or common confusion.

State the domain of 1/(x - 3) and explain why the excluded value stays excluded after any simplification.
Give one relation that is a function but not one-to-one, and explain which property fails.
Use a small sign table to decide where (x - 1)(x + 2) > 0. Check the endpoints separately.
A tank’s volume is measured in litres and time in minutes. What are the units and interpretation of dV/dt? Give an example of a positive and negative value.

A complete mathematical answer names its assumptions and the set of values for which it applies. That is not ceremony: it is the difference between a calculation and a valid statement.

Four retrieval checks

Give a counterexample to the claim “if a number is even, then it is divisible by four” and explain why one example settles it.
State the domain and range of g(x) = √(x - 2).
Explain why cancelling a factor can hide an excluded value.
A tank’s water height is measured in centimetres as a function of minutes. What are the units of its derivative, and what would a negative derivative mean?