CARTESIAN PRODUCTS · RELATIONS · FUNCTIONS · FUNCTION TYPES · ALGEBRA OF FUNCTIONS
A × B = { (a, b) : a ∈ A, b ∈ B }. Every element of A is paired, in order, with every element of B. If n(A) = p and n(B) = q, then n(A × B) = p × q. Type your own sets (comma-separated) and watch the pairing grid build itself.
A relation R from A to B is any subset of A × B, usually described by a rule linking x ∈ A to y ∈ B. Pick two number sets and a rule — the arrow diagram, domain, range and codomain update live.
A relation f from A to B is a function only if every element of A has exactly one image in B. Reuse the arrow-diagram idea: any element sending out two or more arrows breaks the rule (shown in red). Any element of A with zero arrows also fails, since the domain of f must be all of A.
Feed a number x into f(x) = x + 1 and watch its image drop out on the other side.
Pick a function family below to see its graph, domain and range, exactly as introduced in §2.4.1 of the chapter.
For f, g : X → R we can build (f+g)(x) = f(x)+g(x), (f−g)(x), (f·g)(x) = f(x)·g(x), and (f/g)(x) = f(x)/g(x) wherever g(x) ≠ 0. Choose f and g and compare all four combinations on one graph.