Relations and functions – Cartesian product, arrow diagrams, one-one and onto
MathematicsFunctions & GraphsAges 15–16
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Choose two small sets A and B and see A × B as a lattice of ordered pairs; tap pairs (or draw arrows) to build a relation shown at once as a graph, an arrow diagram, a set of ordered pairs and a table, with domain, codomain and range. The sim tests whether the relation is a function and whether the function is one-one, onto or bijective, and on a single set it tests the reflexive, symmetric and transitive properties with a 0–1 matrix and shows the classes of an equivalence relation.
Lesson: Relations and functions: Cartesian product, domain, codomain and range, injective, surjective and bijective functions, equivalence relations
What it shows
The Cartesian product A × B is the set of all ordered pairs (a, b) with a in A and b in B, so it has n(A)·n(B) elements, and every subset of it is a relation from A to B. A relation is a function from A to B when every element of A has exactly one image. A function is one-one when different elements have different images, onto when its range equals the codomain B, and bijective when it is both. A relation on one set is an equivalence relation when it is reflexive, symmetric and transitive; it then splits the set into disjoint equivalence classes.
How to use
In Functions A → B, pick Set A, Set B and a Rule, or tap points of the graph and drag arrows from A to B; the readout and the table show the domain, range and the function test. Random function and Random relation give new examples. In Relations on a set, tap cells of the matrix or link the nodes. Press Start task, build what is asked, then press Check task. Quiz asks whether a relation is a function and which kind.
Parameters you can change
- Screen Functions A → B, Relations on a set, Quiz
- Set A {1, 2, 3}, {1, 2, 3, 4}, {−2, −1, 0, 1, 2}, {a, b, c}
- Set B {1, 2, 3, 4}, {1, 2, 3}, {0, 1, 2, 3, 4}, {1, 2, 3, 4, 5, 6}, {p, q, r, s}
- Rule for A → B None (tap the pairs), y = x + 1, y = 2x, y = x², y = |x|, x < y, x divides y, x + y = 5
- Set for relations on a set {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6}
- Rule on the set None (tap the pairs), x = y, x ≤ y, x < y, x divides y, x ≡ y (mod 2), x ≡ y (mod 3), |x − y| ≤ 1
Questions to explore
- Why is a relation in which one element of A has two images not a function?
- Can a function from a set with 3 elements to a set with 4 elements be onto? Why?
- Which pairs must you add to a relation on {1, 2, 3, 4} to make it an equivalence relation with two classes?
