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Chatty deep dive · Maths 1 Week 1

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Chatty deep dive · Maths 1 Week 1

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2026-08-16

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Now · Sets, Number Systems, Relations & Functions

Mathematics 1 — Week 1

Sets, Number Systems, Relations & Functions

The goal isn't merely to remember definitions. By the end, you should be able to look at a question and think:

"Ah, this is that pattern."

0. The Week 1 Map

Think of the week as four layers:

\boxed{\text{Number Systems}} \rightarrow \boxed{\text{Sets}} \rightarrow \boxed{\text{Relations}} \rightarrow \boxed{\text{Functions}}

And there is an important conceptual progression:

Numbers are objects → sets collect objects → relations connect objects → functions impose a special kind of connection.

That's the backbone.

1. Number Systems

Before sets, we need to know what kinds of numbers we're dealing with.

1.1 Natural numbers

Usually:

\mathbb N=\{1,2,3,\ldots\}

Some texts include

0

\mathbb N_0=\{0,1,2,3,\ldots\}

So always pay attention to the convention being used.

1.2 Integers

\mathbb Z=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}

Contains:

negative integers
zero
positive integers

1.3 Rational numbers

A rational number can be written as:

\frac pq

where

p,q\in\mathbb Z,\qquad q\neq0

Examples:

\frac12,\quad -\frac73,\quad 5,\quad 0.25

Notice that integers are also rational:

5=\frac51

So:

\mathbb Z\subseteq\mathbb Q

1.4 Irrational numbers

Numbers that cannot be expressed as

\frac pq

for integers

p,q

with

q\neq0

Examples:

\sqrt2,\quad \pi,\quad e

Their decimal representations are non-terminating and non-repeating.

1.5 Real numbers

Real numbers contain both rational and irrational numbers:

\mathbb R=\mathbb Q\cup\{\text{irrational numbers}\}

The containment hierarchy is therefore:

\boxed{ \mathbb N\subseteq\mathbb Z\subseteq\mathbb Q\subseteq\mathbb R }

This little diagram is very high-value.

Recognition pattern

If a question asks:

Which of these numbers belongs to $\mathbb Q$ ?

ask:

Can I express it as $\frac pq$ ?

If yes → rational.

2. Set Theory

Now we move from individual objects to collections of objects.

2.1 What is a set?

A set is a collection of distinct objects.

Example:

A=\{1,2,3,4\}

The objects are called elements or members.

We write:

2\in A

meaning:

$2$ is an element of $A$ .

And:

7\notin A

meaning:

$7$ is not an element of $A$ .

3. Set Representation

There are two major forms you'll encounter.

3.1 Roster / listing form

Explicitly list elements:

A=\{2,4,6,8\}

3.2 Set-builder form

Describe a property:

A=\{x\in\mathbb N:x<5\}

Read it as:

The set of all natural numbers $x$ such that $x<5$ .

So:

A=\{1,2,3,4\}

Pattern

Set-builder → roster

Translate the condition into actual elements.

4. Empty Set

The set containing no elements is:

\varnothing

or:

\{\}

Important distinction:

\varnothing\neq\{\varnothing\}

Why?

$\varnothing$

Contains zero elements.

$\{\varnothing\}$

Contains one element.

That one element happens to be the empty set.

This is a classic conceptual trap.

5. Cardinality

Cardinality means:

How many elements does the set contain?

Notation:

|A|

Example:

A=\{2,4,6,8\}

Then:

|A|=4

For the empty set:

|\varnothing|=0

But:

|\{\varnothing\}|=1

6. Subsets

This is one of the most important Week 1 concepts.

Suppose:

A=\{1,2,3\}

and:

B=\{1,2\}

Every element of

B

is also in

A

Therefore:

B\subseteq A

Read:

$B$ is a subset of $A$ .

6.1 Proper subset

If:

B\subseteq A

and

B\neq A

then

B

is a proper subset of

A

Often written:

B\subset A

depending on the notation convention.

6.2 Important pattern

To determine whether:

B\subseteq A

don't ask:

"Does $B$ look smaller?"

Ask:

Is every element of $B$ contained in $A$ ?

Example:

A=\{1,2,3,4\}

B=\{2,4\}

Yes:

B\subseteq A

But:

C=\{2,5\}

No, because:

5\notin A

Therefore:

C\not\subseteq A

7. Set Equality

Two sets are equal if they contain exactly the same elements.

Example:

A=\{1,2,3\}

B=\{3,2,1\}

Then:

A=B

because order doesn't matter in a set.

Likewise:

\{1,1,2,2,3\}=\{1,2,3\}

because duplicate elements don't create new set elements.

Contrast

A list:


text
[1, 2, 3]

and


text
[3, 2, 1]

may be different.

A set:

\{1,2,3\}=\{3,2,1\}

8. Power Set

The power set of

A

is the set containing all subsets of $A$ .

Notation:

\mathcal P(A)

Suppose:

A=\{a,b\}

Its subsets are:

\varnothing,\{a\},\{b\},\{a,b\}

Therefore:

\mathcal P(A) = \{\varnothing,\{a\},\{b\},\{a,b\}\}

If:

|A|=n

then:

\boxed{|\mathcal P(A)|=2^n}

Why?

Each element has two possibilities:

included / not included.

For

n

elements:

2\times2\times\cdots\times2=2^n

This is one of those patterns that later connects directly to counting in Statistics.

9. Set Operations

Now the fun part.

Suppose:

A=\{1,2,3,4\}

B=\{3,4,5,6\}

9.1 Union

Union means:

Everything in $A$ or $B$ .

A\cup B

Therefore:

A\cup B=\{1,2,3,4,5,6\}

Mental model

\boxed{\text{OR}}

10. Intersection

Intersection means:

Things common to both.

A\cap B

Therefore:

A\cap B=\{3,4\}

Mental model

\boxed{\text{AND}}

This OR/AND interpretation is extremely useful.

11. Difference

A-B

means:

Elements in $A$ that are not in $B$ .

Therefore:

A-B=\{1,2\}

Notice:

A-B\neq B-A

Here:

B-A=\{5,6\}

So set difference is not commutative.

12. Complement

If the universal set is

U

, then the complement of

A

is everything in

U

that isn't in

A

A^c=U-A

Example:

U=\{1,2,3,4,5,6\}

A=\{1,2,3\}

Then:

A^c=\{4,5,6\}

Important

You cannot determine a complement without knowing the universe.

13. The Core Set Pattern

Given:

A=\{1,2,3,4\}

B=\{3,4,5,6\}

Then:

Expression	Result	Interpretation
$A\cup B$	$\{1,2,3,4,5,6\}$	OR
$A\cap B$	$\{3,4\}$	AND
$A-B$	$\{1,2\}$	A but not B
$B-A$	$\{5,6\}$	B but not A

This is a pattern, not four facts to memorize.

14. Set Identities

Some useful identities:

Commutativity

A\cup B=B\cup A

A\cap B=B\cap A

But:

A-B\neq B-A

in general.

Associativity

(A\cup B)\cup C=A\cup(B\cup C)

(A\cap B)\cap C=A\cap(B\cap C)

Distributivity

A\cap(B\cup C) = (A\cap B)\cup(A\cap C)

and:

A\cup(B\cap C) = (A\cup B)\cap(A\cup C)

15. De Morgan's Laws

Very high-value.

\boxed{(A\cup B)^c=A^c\cap B^c}

and:

\boxed{(A\cap B)^c=A^c\cup B^c}

Mental translation:

NOT (A OR B) = NOT A AND NOT B

NOT (A AND B) = NOT A OR NOT B

This same logic appears later in:

Statistics
CT
Python conditionals

So this isn't just a Maths 1 trick.

16. Relations

Now we move from collections to connections.

Suppose:

A=\{1,2,3\}

B=\{a,b\}

The Cartesian product is:

A\times B

which contains ordered pairs:

A\times B= \{ (1,a),(1,b), (2,a),(2,b), (3,a),(3,b) \}

In general:

|A\times B|=|A||B|

if both sets are finite.

17. Why Ordered Pairs Matter

Notice:

(1,a)\neq(a,1)

because order matters.

This is different from sets:

\{1,a\}=\{a,1\}

So:

Set: order irrelevant
Ordered pair: order matters

That's a common exam trap.

18. Relation

A relation from

A

B

is essentially a subset of $A\times B$ .

Suppose:

R=\{(1,a),(2,b),(3,a)\}

Then

R

is a relation from

A

B

Think:


text
A                  B

1 ───────────────→ a
2 ───────────────→ b
3 ───────────────→ a

A relation simply specifies which connections exist.

19. Function

Now comes the crucial restriction.

A function from

A

B

is a relation where:

Every input has exactly one output.

Notation:

f:A\to B

Think:


text
input → exactly one output

20. Function Recognition Pattern

Suppose:


text
1 → a
2 → b
3 → a

Valid function.

Why?

Each input has exactly one arrow leaving it.

It is perfectly okay for multiple inputs to have the same output.

But:


text
1 → a
1 → b
2 → a

is not a function.

Why?

Input

1

has two outputs.

And:


text
1 → a
2 → b
3

is not a function from the entire domain $\{1,2,3\}$ , because

3

has no output.

Golden rule

\boxed{\text{Every input → exactly one output}}

21. Domain, Codomain & Range

Suppose:

f:A\to B

Then:

Domain

The set of allowed inputs:

A

Codomain

The set in which outputs are supposed to lie:

B

Range

The outputs that are actually produced.

These are not necessarily the same.

Example:

f:\{1,2,3\}\to\{a,b,c,d\}

with:

1\to a

2\to b

3\to a

Then:

\text{Domain}=\{1,2,3\}

\text{Codomain}=\{a,b,c,d\}

but:

\text{Range}=\{a,b\}

This distinction is very testable.

22. One-to-One / Injective

A function is one-to-one if:

Different inputs always produce different outputs.

Formally:

f(a)=f(b)\Rightarrow a=b

Example:

1\to a

2\to b

3\to c

is injective.

But:

1\to a

2\to a

is not injective.

Mental model

No two inputs collide.

23. Onto / Surjective

A function is onto if:

Every element of the codomain is actually hit.

Suppose:

A=\{1,2,3\}

B=\{a,b,c\}

and:

1\to a,\quad2\to b,\quad3\to c

Then every codomain element gets used.

So:

f:A\to B

is onto.

24. Bijective

A function is bijective if it is both:

\boxed{\text{injective + surjective}}

So:

no collisions + nothing left unused.

This is the strongest one-to-one correspondence between two sets.

25. The Function Taxonomy

Keep this mental picture:


text
                    FUNCTIONS
                       │
             ┌─────────┴─────────┐
             │                   │
        Injective?           Surjective?
             │                   │
             └─────────┬─────────┘
                       │
                  Both = Bijective

And remember:

Injective: no output collision.
Surjective: no unused codomain element.
Bijective: both.

26. Week 1 Pattern Bank — First Version

We're going to maintain this and expand it as we study.

ID	Pattern	Difficulty
W1-N1	Identify number-system membership	Easy
W1-S1	Set membership	Easy
W1-S2	Set-builder ↔ roster	Easy
W1-S3	Cardinality	Easy
W1-S4	Subset checking	Easy
W1-S5	Power-set cardinality	Easy
W1-S6	Union/intersection/difference	Easy
W1-S7	Multi-operation set expression	Medium
W1-S8	Complement	Easy
W1-S9	Set identities	Medium
W1-S10	De Morgan transformations	Medium
W1-R1	Cartesian product	Easy
W1-R2	Relation identification	Easy
W1-F1	Function identification	Easy
W1-F2	Domain/codomain/range	Easy
W1-F3	Injective checking	Medium
W1-F4	Surjective checking	Medium
W1-F5	Bijective checking	Medium
W1-F6	Mixed representation	Hard

27. Now I want you to do something

Don't just read all that and go:

"yeah yeah makes sense"

That's the exact failure mode we're trying to avoid.

Let's do a diagnostic round before I give you the next layer.

No calculator. Don't search. Don't overthink.

Diagnostic — Week 1

Q1 — Number systems

Classify each as natural, integer, rational, irrational, real where applicable:

-7,\quad 0,\quad \frac34,\quad \sqrt2,\quad \pi

Q2 — Sets

Let:

A=\{1,2,3,4,5\}

B=\{2,4,6\}

Find:

A\cap B

A\cup B

A-B

Q3 — Subsets

Is:

\{1,3\}\subseteq\{1,2,3,4\}?

Is:

\{1,5\}\subseteq\{1,2,3,4\}?

Give the reason, not merely yes/no.

Q4 — Power set

If:

|A|=5

what is:

|\mathcal P(A)|?

And why?

Q5 — Cartesian product

If:

A=\{1,2\}

and:

B=\{x,y,z\}

what is:

|A\times B|?

Q6 — Function recognition

Is this a function?


text
1 → a
2 → b
3 → a

Why?

Q7 — Function recognition

Is this a function?


text
1 → a
1 → b
2 → c

Why?

Q8 — Domain / codomain / range

Suppose:

f:\{1,2,3\}\to\{a,b,c,d\}

with:

f(1)=a,\qquad f(2)=b,\qquad f(3)=a

Give:

Domain
Codomain
Range
Is it injective?
Is it surjective?
Is it bijective?

Q9 — Slightly nastier

Let:

A=\{1,2,3,4\}

B=\{3,4,5\}

C=\{4,5,6\}

Find:

A\cap(B\cup C)

Do not calculate it mentally in one jump. Show the intermediate set.

Q10 — Pattern recognition

Suppose somebody gives you:

"A function has domain $\{1,2,3,4\}$ and codomain $\{a,b,c\}$ . Each element of the domain maps to exactly one element of the codomain, and every codomain element is used."

What property does the function definitely have?

And what cannot you conclude from this information alone?

Reply with your answers however you naturally would — messy is fine. I care more about your reasoning than formatting.

Then I'll diagnose each answer as:

\boxed{\text{Concept gap / Pattern gap / Execution error / Careless error / Correct}}

and we'll use that to decide whether to push deeper into Week 1 or move to Week 2.