0.1 Course foundations and notation

Why this chapter comes first

The first topic in a mathematics methods course is not a technique but a language. If notation is sloppy, later algebra becomes impossible to audit: you cannot tell what is assumed, what is being proved, or whether a step keeps the same solution set.

This chapter therefore does three jobs at once:

it fixes the meaning of basic set notation;
it introduces the logic symbols that will appear in every proof;
it explains the difference between a legitimate transformation and a step that changes the problem.

Sets and membership

Definition

Set

A set is a collection of distinct objects. The objects inside the set are its elements or members.

If x is an element of a set $A$ , we write $x \in A$ . If x is not an element of $A$ , we write $x \notin A$ .

Set membership is the smallest unit of mathematical language here. Once you can state membership precisely, you can state subsets, domains, ranges, and solution sets precisely as well.

Definition

Subset and equality of sets

Let $A$ and $B$ be sets.

$A \subseteq B$ means every element of $A$ is also an element of $B$ .
$A = B$ means $A$ and $B$ have exactly the same elements.

Equivalently, $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$ .

That equivalence is worth remembering. To prove two sets are equal, it is often cleaner to prove the two inclusions separately.

Worked example

A set written two ways

The set of positive even integers can be written as

\{2m : m \in Z^+\}

or as

\{n \in Z^+ : n = 2m \text{ for some } m \in Z^+\}.

The first form emphasizes the pattern; the second form emphasizes the property that defines the elements.

Set-builder notation and standard sets

The course uses the notation

\{x \in S : P(x)\}

for the set of all elements x in a universe $S$ that satisfy the property P(x).

This is not just shorthand. It tells you three things at once:

where the objects are coming from;
what property selects them;
whether the property is being imposed on a larger ambient set.

Definition

Common number sets

$N := \{0,1,2,3,\dots\}$ is the set of natural numbers.
$Z := \{\dots,-2,-1,0,1,2,\dots\}$ is the set of integers.
$Z^+ := \{1,2,3,\dots\}$ is the set of positive integers.
Q := \{p/q : p,q \in Z, \ q \neq 0\} is the set of rational numbers.
$R$ is the set of real numbers.

The notation $:=$ means “is defined to be.” It is used when the symbol on the left is being introduced rather than merely asserted.

Worked example

From a property to an interval

The set

\{x \in R : x^2 > 1\}

is the same as

(-\infty,-1) \cup (1,\infty).

Reason: $x^2 > 1$ means $|x| > 1$ , so x must lie outside the closed interval $[-1,1]$ .

Common mistake

Do not confuse set subtraction with division

Write $R \setminus \{2\}$ for the set of all real numbers except 2. The expression R \2 is not valid notation because 2 is not a set.

Set operations and interval notation

Definition

Intersection, union, and difference

Let $A$ and $B$ be sets.

$A \cap B$ contains the elements common to both sets.
$A \cup B$ contains the elements that belong to at least one of the sets.
$A \setminus B$ contains the elements of $A$ that are not in $B$ .

These operations appear constantly when you describe solution sets, domains, and the outcome of splitting a problem into cases.

The interval symbols are shorthand for subsets of $R$ :

$[a,b] = \{x \in R : a \le x \le b\}$ ;
$(a,b) = \{x \in R : a < x < b\}$ ;
$(a,b] = \{x \in R : a < x \le b\}$ ;
$[a,\infty) = \{x \in R : a \le x\}$ .

The closed bracket means the endpoint is included; the open bracket means it is excluded.

Worked example

Two ways to describe the same set

The set of real numbers whose square exceeds 1 can be written in at least three equivalent ways:

\{x \in R : x^2 > 1\} = (-\infty,-1) \cup (1,\infty) = R \setminus [-1,1].

All three descriptions are useful, but they emphasize different aspects of the same set.

Logic symbols and quantifiers

The course uses a small number of logic symbols repeatedly:

$\forall$ means “for all” or “for every”;
$\exists$ means “there exists at least one”;
$\exists !$ means “there exists exactly one”;
$\Rightarrow$ means “implies”;
$\Leftrightarrow$ means “if and only if.”

Definition

A fully stated mathematical claim

A serious mathematical statement should make clear:

the domain of the variables;
the operation being performed;
the conclusion being claimed.

The order of quantifiers matters. These two statements are not the same:

\forall x \in Z, \exists y \in Z, x+y=0

and

\exists y \in Z, \forall x \in Z, x+y=0.

The first is true: for each integer x, choose $y=-x$ . The second is false: no single integer y cancels every integer x at once.

Worked example

Reading quantifiers carefully

The statement

\exists ! x \in R \text{ such that } x^2 - 2x + 1 = 0

means that the equation has exactly one real solution. Since

x^2 - 2x + 1 = (x-1)^2,

the unique solution is $x=1$ .

Common mistake

Do not swap the order of quantifiers casually

$\forall x \exists y$ and $\exists y \forall x$ usually mean different things. If you change the order, you usually change the statement itself.

Functions, domain, codomain, and range

Definition

Function

A function is a rule that assigns to each element of a domain exactly one element of a codomain.

If $b = f(a)$ , then b is the value of f at a, or the image of a. The range of f, written f(A) or range(f), is the set of all outputs that actually occur.

The domain is not optional decoration. It tells you where the rule is legally defined. The codomain tells you where the outputs are allowed to live. The range is the subset of the codomain that the function actually reaches.

Worked example

Same formula, different range

Consider $f(x)=x^2$ .

If $f : R \to R$ , then $range(f) = [0,\infty)$ .
If $f : Z^+ \to Z^+$ , then range(f) is the set of perfect squares in $Z^+$ .

The formula is the same, but the domain changes the meaning of the function.

Common mistake

Range is not the same as codomain

If a function is declared as $f : A \to B$ , then $B$ is the codomain, not necessarily the range. The range is usually smaller unless the function is onto.

Baseline transformation rules

Some steps preserve an equation exactly. Others only make sense when extra domain information is known.

Definition

Equality-preserving and domain-sensitive steps

Equality-preserving steps: adding or subtracting the same term on both sides, multiplying by a nonzero constant, or replacing an expression by an equivalent one.
Domain-sensitive steps: dividing by an expression that may be zero, squaring both sides, taking square roots, and applying inverse functions without checking their domain restrictions.

The main habit in this course is simple: every time you transform a statement, ask whether the new statement has exactly the same solution set as the old one.

That habit is why the course keeps repeating words like domain, legal, equivalent, and verify.

Equations, identities, and conditions

One reason early notation matters so much is that an equation and an identity ask for different kinds of work.

Definition

Equation versus identity

An equation asks for the values in the allowed domain that make two expressions equal.

An identity claims that two expressions are equal for every value in a specified domain.

For example,

(x-1)(x+1) = x^2 - 1

is an identity in $R$ , because both sides agree for every real x.

By contrast,

x^2 - 1 = 0

is an equation. We are not claiming it is always true. We are asking for the values of x that make it true, namely $x = 1$ or $x = -1$ .

This distinction is exactly why the course keeps talking about preserving the solution set. When you solve an equation, every legal step must keep the same set of valid values. When you verify an identity, every step must remain valid for the whole stated domain.

Absolute value and distance notation

Absolute value is another place where careful notation matters early.

Definition

Absolute value

For a real number x,

|x| = \begin{cases} x, & x \ge 0, \\ -x, & x < 0. \end{cases}

So |x| is the distance from x to 0 on the number line.

More generally, $|a-b|$ measures the distance between a and b. That is why absolute-value inequalities often translate directly into intervals.

Worked example

Rewrite an absolute-value inequality as an interval

Solve

|x-3| < 2.

The statement says that the distance from x to 3 is less than 2, so x must lie within 2 units of 3. Therefore

1 < x < 5.

Written in interval form, the solution set is (1,5).

Short checklist before you move on

Before you start a calculation, make sure you can answer these questions:

What is the variable domain?
Is every operation legal on that domain?
Are you solving an equation, proving an identity, or describing a set?
If you split into cases, have you covered every case exactly once?

Quick checks

Quick check

Why must $x \neq 0$ be stated before dividing an equation by `x`?

Think about both legality and solution sets.

Solution

Answer

Quick check

Which notation shows that $A$ and $B$ have the same elements: $A \subseteq B$ or $A = B$ ?

Use the definition of set equality.

Solution

Answer

Quick check

Read the statement: $\forall x \in Z, \exists y \in Z, x+y=0$ . What does it say in words?

Pay attention to the order of the quantifiers.

Solution

Answer

Quick check

If $f : R \to R$ is given by $f(x)=x^2$ , what is its range?

Do not confuse the range with the codomain.

Solution

Answer

Exercises

Rewrite the set of all positive odd integers in set-builder notation.
Decide whether $\{x \in R : x^2 \le 4\}$ is an interval or a union of intervals, and write it explicitly.
Explain why $\exists ! x \in R$ is stronger than $\exists x \in R$ .
State the domain, codomain, and range of f(x)=1/(x-1) if $f : R \setminus \{1\} \to R$ .

Use the quick-check answers above as a model for the level of precision the course expects.

0.1 Course foundations and notation

MATH1025: Preparatory mathematics

Why this chapter comes first

Sets and membership

Set

Subset and equality of sets

A set written two ways

Set-builder notation and standard sets

Common number sets

From a property to an interval

Do not confuse set subtraction with division

Set operations and interval notation

Intersection, union, and difference

Two ways to describe the same set

Logic symbols and quantifiers

A fully stated mathematical claim

Reading quantifiers carefully

Do not swap the order of quantifiers casually

Functions, domain, codomain, and range

Function

Same formula, different range

Range is not the same as codomain

Baseline transformation rules

Equality-preserving and domain-sensitive steps

Equations, identities, and conditions

Equation versus identity

Absolute value and distance notation

Absolute value

Rewrite an absolute-value inequality as an interval

Short checklist before you move on

Quick checks

Why must x≠0x \neq 0x=0 be stated before dividing an equation by x?

Answer

Which notation shows that AAA and BBB have the same elements: A⊆BA \subseteq BA⊆B or A=BA = BA=B?

Answer

Read the statement: ∀x∈Z,∃y∈Z,x+y=0\forall x \in Z, \exists y \in Z, x+y=0∀x∈Z,∃y∈Z,x+y=0. What does it say in words?

Answer

If f:R→Rf : R \to Rf:R→R is given by f(x)=x2f(x)=x^2f(x)=x2, what is its range?

Answer

Exercises

Prerequisites

Key terms in this unit

More notes in this series

Why must $x \neq 0$ be stated before dividing an equation by `x`?

Which notation shows that $A$ and $B$ have the same elements: $A \subseteq B$ or $A = B$ ?

Read the statement: $\forall x \in Z, \exists y \in Z, x+y=0$ . What does it say in words?

If $f : R \to R$ is given by $f(x)=x^2$ , what is its range?