A relation between two sets A and B is just a collection of ordered pairs. You can write it as a list — \{(1, p), (2, q), (3, s)\} — or you can draw it. The arrow diagram is the drawing: put A on the left, B on the right, and draw one arrow from a to b for each pair (a, b) in the relation. Nothing else. The picture is the relation.
The picture the definition wants you to draw
Take A = \{1, 2, 3, 4\} on the left and B = \{p, q, r, s\} on the right. Each arrow you draw from an element of A to an element of B adds exactly one pair to the relation. No arrow means that pair is absent.
How to read an arrow
Every arrow has a starting dot in A and an ending dot in B. If an arrow runs from 2 to q, then the pair (2, q) is in the relation. If no arrow runs between two specific dots, that pair is not in the relation — the absence of an arrow is just as meaningful as its presence.
Three conventions worth fixing in your head:
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Arrows are directional. An arrow from 1 to p is the pair (1, p). If both (1, p) and (p, 1) were in a relation on A only (i.e., a relation on one set), you would draw two arrows — one in each direction. For a relation from A to B with distinct sets, the direction is always left to right, and (p, 1) is not even a valid pair (since p \notin A).
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Multiple arrows from one element are allowed. Element 1 on the left can have arrows to p and to r. That simply means both pairs (1, p) and (1, r) are in the relation. A relation does not require uniqueness — only a function does.
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An element with no arrow is still in its set. If element 3 has no outgoing arrow, that means 3 \notin \text{dom}(R). Element 3 still exists in A; it is just unrelated to anything.
What the arrow count tells you
The number of arrows is exactly the number of pairs in the relation, |R|. So:
| Arrow count | Relation |
|---|---|
| 0 | Empty relation |
| $ | A |
| $ | A |
The maximum possible arrow count is |A| \cdot |B| — there are only so many pairs you can draw between two finite sets. In the figure above, |A| = |B| = 4, so at most 16 arrows are possible, which is why the slider tops out at 16.
Why: the arrow diagram is a literal picture of the definition "a relation is a subset of A \times B." Each arrow is one ordered pair. The set of arrows is the relation. Nothing is abstracted away, nothing is compressed; the definition walks onto the page.
Arrow diagram vs. grid diagram
The same relation can be drawn two ways: as arrows between columns, or as filled cells on a grid. The two views capture the same information, but each highlights a different property.
Arrow diagram. Best for reading the domain (which dots on the left have any arrow leaving them) and the range (which dots on the right have any arrow arriving). Also best for seeing when something looks "function-shaped" — a function has exactly one arrow from each left-dot.
Grid diagram. Best for counting pairs quickly (it is just a count of filled cells) and for seeing symmetry (for a relation on A to itself, the filled cells form a pattern that is mirrored across the main diagonal iff the relation is symmetric).
For relations from A to B (two distinct sets), the arrow diagram is usually clearer. For relations on a single set A, the directed-graph picture — nodes for each element, arrows between them — is the natural choice, and it is a close cousin of the arrow diagram.
Edge cases that test your reading
Two arrows from the same element. Element 1 has arrows to both p and r. This means (1, p) \in R and (1, r) \in R. It does not violate anything — relations do not require the left element to have a unique partner. It would violate the function property if R were claimed to be a function, but it is not.
An isolated element on the right. Element r has no incoming arrows. That means r is not in the range of R. It still belongs to the set B; it just happens not to be hit by the relation.
An arrow from an element to itself. Only possible when A = B — a self-loop indicates the pair (a, a). In the two-column arrow diagram with A \neq B, self-loops do not appear; in directed-graph pictures of relations on one set, they do.
Three sample relations on the same sets
Set A = \{1, 2, 3, 4\}, B = \{p, q, r, s\}.
Relation 1 — "map each number to the letter at that position": R_1 = \{(1, p), (2, q), (3, r), (4, s)\}. Four arrows, each going straight across. This is a function (one arrow per left-dot) and in fact a bijection.
Relation 2 — "map each number to every letter before its position": R_2 = \{(2, p), (3, p), (3, q), (4, p), (4, q), (4, r)\}. Six arrows. Not a function — element 3 has two outgoing arrows.
Relation 3 — "map each number to every letter at or after the vowel q": R_3 = \{(1, q), (1, r), (1, s), (2, q), (2, r), (2, s), (3, q), (3, r), (3, s), (4, q), (4, r), (4, s)\}. Twelve arrows. Every left-dot has three outgoing arrows. Not a function.
Each of these is a valid relation; the arrow diagrams differ entirely in look, but each faithfully draws its corresponding pair-set.
The mental move
When a problem writes out a relation as a list of pairs, draw the arrow diagram. When it draws an arrow diagram, read off the pair-list. These two formats are perfectly interchangeable, and becoming fluent at flipping between them is half the work of becoming fluent at relations.
Arrows are pairs. Pairs are arrows. The picture is the relation — not a schematic, not a hint, but the thing itself.
Related: Relations · Ordered-Pair Plotter: Pick a Relation, See It on the Grid · Symmetry Check: Missing Reverse Arrows Glow Red · Set Operations