Foundations — Domain, codomain, range

Before you can read the parent note confidently, you must own every squiggle it uses. This page starts from literally nothing and earns each symbol one at a time. If a symbol below feels obvious, good — read it anyway, because the parent note leans on it.

1. What is a set?

Picture a paper bag with objects dropped in. Order does not matter, and repeats do not count — a bag with two identical apples is the same as a bag with one apple.

$\{1,2,3\}\{\text{apple},\text{orange}\}\{\}\text{ (the empty bag)}$

Why the topic needs it: domain, codomain and range are all sets. Every idea on this topic is "which things are in which bag."

2. The symbol $\in$ — "is inside"

Look at figure s01: the arrow poking into the bag is exactly what $\in$ means.

Why the topic needs it: the parent writes "for each $x\in A$" — meaning "for every input $x$ living inside the domain bag $A$." Without $\in$ you cannot even state what a function does.

3. The number bags: $\mathbb{R}$ and friends

Picture an endless ruler stretching left and right forever. Every dot on it is one element of $\mathbb{R}$.

Why the topic needs it: the parent's functions are declared like $f:\mathbb{R}\to \mathbb{R}$ — the input bag is the number line. You must see $\mathbb{R}$ as "the whole line" to picture domains and ranges as pieces of that line.

4. Intervals: naming a piece of the line

We rarely use the whole line. We chop out a stretch of it and name it.

Picture a fence around a stretch of ruler. A solid dot ● means the endpoint is inside; a hollow dot ○ means it is not.

Why the topic needs it: answers like "Range $=[0,25]$" are intervals. If you cannot decode a bracket you cannot state a single answer on this topic.

5. Combining and removing: $\cup$ and $\setminus$

Picture the number line with a single point scooped out, leaving a tiny gap.

Why the topic needs it: the parent's Example 3 has domain $\mathbb{R}\setminus \{2\}$ (division by zero forbidden) and range $\mathbb{R}\setminus \{1\}$ (one value never comes out). These holes are the whole point of that example.

6. $\subseteq$ — "is a smaller bag inside a bigger bag"

Picture a small circle drawn completely inside a big circle.

Why the topic needs it: the single most important fact — "range $\subseteq$ codomain, equal only when the function is onto" — is written in this symbol. This directly feeds Surjective Functions.

7. The map arrow $\to$ and the function name $f$

Picture a machine with a funnel on the left labelled $A$ and a bin on the right labelled $B$.

Why the topic needs it: every function in the parent is announced this way. The arrow tells you the domain and codomain before you compute anything — half the answer is handed to you by the notation.

8. The "there exists" symbol $\exists$

Picture a searchlight sweeping the domain bag, hunting for one input that hits a chosen output. If it finds even one, the search succeeds.

Why the topic needs it: finding a range is exactly asking "for which $y$ does an input exist?" The parent's whole range-algorithm is this symbol turned into steps.

Prerequisite map

Read it top-down: bags and the "inside" symbol are the roots; from them grow number-lines, intervals, subsets, the map arrow, and the searchlight — all of which pour into the topic at the bottom.

Equipment checklist

Cover the answers and test yourself. If any line stumps you, re-read its section above before touching the parent note.

Connections

• Domain, codomain, range — the parent topic every symbol here serves.
• Function Basics — builds directly on the map arrow $f:A\to B$.
• Graphing Functions — intervals here become the x- and y-axis extents there.
• Surjective Functions — the $\subseteq$ fact becomes "onto = range equals codomain."
• Inverse Functions — needs the range/codomain distinction defined here.
• Composite Functions — uses $\subseteq$ to fit one function's range into another's domain.
• Injective Functions — reasons about outputs, i.e. elements of the range.