Understanding Relations in Mathematics (With Grid Examples and Key Properties)
- Tyler Buffone
- Sep 28
- 4 min read
Updated: Oct 9
Background
Sets and Ordered Pairs
A set is a collection of things, for example A = {1, 2, 3}.
An ordered pair (a, b) records a first thing and a second thing. Order matters: (a, b) ≠ (b, a) in general.
Cartesian Products
A × B is the set of all ordered pairs where the first entry is from A and the second entry is from B.
For example, if A = {1, 2} and B = {♡, ♢}, then
A × B = {(1, ♡), (1, ♢), (2, ♡), (2, ♢)}.
A Picture that Makes Relations Easy
Grid View: Imagine a table with rows labelled by elements of A and columns by elements of B.
For instance, if we let A = {1, 2, 3, 4} and B = {♡, ♢, ♧, ♤}, then we can construct the following table that pictures the entire Cartesian product A × B.
| ♡ | ♢ | ♧ | ♤ |
1 | (1, ♡) | (1, ♢) | (1, ♧) | (1, ♤) |
2 | (2, ♡) | (2, ♢) | (2, ♧) | (2, ♤) |
3 | (3, ♡) | (3, ♢) | (3, ♧) | (3, ♤) |
4 | (4, ♡) | (4, ♢) | (4, ♧) | (4, ♤) |
If A = B, the diagonal pairs are (a, a) that run from the top-left to bottom-right of the grid.
For instance, if we let A = {1, 2, 3, 4}, then we can construct the following table that pictures the entire Cartesian product A × A.
| 1 | 2 | 3 | 4 |
1 | (1, 1) | (1, 2) | (1, 3) | (1, 4) |
2 | (2, 1) | (2, 2) | (2, 3) | (2, 4) |
3 | (3, 1) | (3, 2) | (3, 3) | (3, 4) |
4 | (4, 1) | (4, 2) | (4, 3) | (4, 4) |
What is a Relation?
A relation R from A to B is simply some subset of A × B.
We write aRb when (a, b) ∈ R
If A = B, then R is a relation on A (all pairs live in A × A).
Essentially, a relation is a yes/no rule about ordered pairs.
Is (a, b) in the chosen subset?
If yes, then aRb.
If no, then they are not related.
The Properties to Test
There are some standard properties of relations on sets; there are two contexts with respect to these.
Relations from A to B (possibly different sets)
Left-total: every a ∈ A relates to at least one b ∈ B.
Grid view: every row has at least one checkmark.
Functional: no a ∈ A relates to two different values of b ∈ B.
Grid view: every row has at most one checkmark.
Together, left-total + functional means the relation behaves like a function A → B.
Functional but not left-total means a partial function.
Example 1: Left-Total Relation (not functional)
Consider the two sets A = {1, 2, 3, 4} and B = {♡, ♢, ♧, ♤}.
Suppose we define the following relation from A to B:
R = {(1, ♡), (1, ♢), (2, ♧), (3, ♤), (4, ♡)}.
Let's observe this using the grid view:
| ♡ | ♢ | ♧ | ♤ |
1 | (1, ♡) ✓ | (1, ♢) ✓ | (1, ♧) ✗ | (1, ♤) ✗ |
2 | (2, ♡) ✗ | (2, ♢) ✗ | (2, ♧) ✓ | (2, ♤) ✗ |
3 | (3, ♡) ✗ | (3, ♢) ✗ | (3, ♧) ✗ | (3, ♤) ✓ |
4 | (4, ♡) ✗ | (4, ♢) ✗ | (4, ♧) ✗ | (4, ♤) ✓ |
Check if left-total:
1 appears with ♡ and ♢,
2 with ♧,
3 with ♤,
4 with ♡.
Every row has something → left-total.
Check if functional:
1 maps to both ♡ and ♢ → not functional.
Relations on a Single Set A (so pairs sit in A × A)
Reflexive: every element relates to itself.
Grid: the diagonal is fully checked.
Irreflexive: no element relates to itself
Grid: the diagonal is completely empty
Note: a relation can be neither reflexive nor irreflexive if some diagonal pairs are in and some are out.
Symmetric: if aRb then bRa
Grid: every checkmark mirrors across the diagonal.
Anti-symmetric: if aRb and bRa then a = b
Grid: you never see a mirrored off-diagonal pair. Diagonal checks are fine.
Important: symmetric and anti-symmetric are not opposites.
Transitive: if aRb and bRc, then aRc.
Total (linear): any two elements are comparable.
For all a, b, at least one of a = b, aRb, or bRa holds.
Example 2: Relation that is irreflexive, symmetric, and not transitive
Consider the set A = {1, 2, 3, 4}.
Suppose we define the following relation from A to A: R = {(1, 2), (2, 1), (2, 3), (3, 2)}.
Let's observe this using the grid view.
| 1 | 2 | 3 | 4 |
1 | (1, 1) ✗ | (1, 2) ✓ | (1, 3) ✗ | (1, 4) ✗ |
2 | (2, 1) ✓ | (2, 2) ✗ | (2, 3) ✓ | (2, 4) ✗ |
3 | (3, 1) ✗ | (3, 2) ✓ | (3, 3) ✗ | (3, 4) ✗ |
4 | (4, 1) ✗ | (4, 2) ✗ | (4, 3) ✗ | (4, 4) ✗ |
Check if irreflexive:
No (a, a) are included (the diagonal is completely empty) → irreflexive.
Check: if symmetric:
Each pair has its mirror: (1, 2) with (2, 1), (2, 3) with (3,2) → symmetric.
Check if transitive:
(1, 2) and (2, 3) are in R, but (1, 3) is not → not transitive.
How Properties Combine into the Big Families
Equivalence relation: reflexive + symmetric + transitive.
These carve the set into equivalence classes that form a partition.
Weak order: reflexive + anti-symmetric + transitive.
A way to rank things, like ≤.
Partial if not total; total (linear) if total.
Strong order: irreflexive + transitive.
The strict version of an order, like <.
Strict partial if not total; strict total (linear) if total.
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