Flash Cards: Relations and Functions

A relation is a set of ordered pairs, where each pair consists of elements from two sets.

The Cartesian product of sets A and B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B, denoted as A × B.

An ordered pair is a pair of elements (a, b) where the order matters, meaning (a, b) is not the same as (b, a).

List all possible combinations of elements from the first set with elements from the second set in ordered pairs.

A × B contains pairs (a, b) with a from A and b from B, while B × A contains pairs (b, a) with b from B and a from A.

A function is a special type of relation where each element in the domain is associated with exactly one element in the codomain.

The domain is the set of all possible input values for the function.

The codomain is the set of all potential output values of a function.

Examples include f(x) = x², f(x) = 3x + 1, and f(x) = sin(x).

The range is the set of all actual output values that a function can produce.

In a function, no two ordered pairs have the same first element with different second elements.

A one-to-one function assigns each element in the domain to a unique element in the codomain.

An onto function covers every element in the codomain; every element of the codomain is the output of some input from the domain.

A function is often denoted as f: A → B, where A is the domain and B is the codomain.

A composite function is formed when one function is applied to the results of another function, denoted as (f ∘ g)(x) = f(g(x)).

Thinking that a relation with repeating first elements is a function; a function must have unique outputs for each input.

If f(x) = x + 2 and g(x) = 3x, then f(g(x)) = f(3x) = 3x + 2.

The image is the set of all output values corresponding to the elements in the domain.