Ordered pairs and Cartesian products
An ordered pair (a, b) has a first element a and a second element b. Order matters: (2, 3) ≠ (3, 2). Two ordered pairs are equal only if their corresponding elements are equal: (x + 1, 4) = (5, y − 2) means x = 4 and y = 6.
The Cartesian product of sets A and B is the set of all ordered pairs with first element from A and second from B:
A × B = {(a, b) : a ∈ A, b ∈ B}
Example. A = {1, 2}, B = {x, y, z}. A × B = {(1, x), (1, y), (1, z), (2, x), (2, y), (2, z)}.
If A has p elements and B has q, then A × B has pq elements.
A × B ≠ B × A in general.
If A or B is empty, A × B is empty.
ℝ × ℝ is the set of all points in the plane, the familiar xy-plane.
Relations
A relation R from A to B is any subset of A × B. If (a, b) ∈ R, we say a is related to b and write a R b.
The set of first elements is the domain of R.
The set of second elements is the range of R.
The whole set B is the codomain.
Example. A = {1, 2, 3, 4} and R = {(a, b) : b = a + 1, a, b ∈ A}. Then R = {(1, 2), (2, 3), (3, 4)}, domain {1, 2, 3} and range {2, 3, 4}.
Since A × B has 2^(pq) subsets, there are 2^(pq) possible relations from A to B.
Functions
A function f from A to B, written f : A → B, is a relation in which every element of A has exactly one image in B.
Two conditions must hold:
every element of A appears as a first element, and
no element of A appears with two different second elements.
Example. For A = {1, 2, 3}:
{(1, 4), (2, 5), (3, 4)} is a function. Two elements having the same image is allowed.
{(1, 4), (1, 5), (2, 6), (3, 7)} is not: 1 has two images.
{(1, 4), (2, 5)} is not a function on A: 3 has no image.
Vertical line test: a graph in the plane is the graph of a function of x if and only if no vertical line meets it more than once. A circle fails the test.
For f : A → B, A is the domain, B the codomain, and the set of actual images {f(x) : x ∈ A} is the range. The range is a subset of the codomain.
Finding the domain and range of real functions
A real function has a domain and range that are subsets of ℝ. When only a formula is given, the domain is the set of all real x for which the formula makes sense.
Example. f(x) = 1/(x − 3). Division by zero is not allowed, so the domain is ℝ − {3}.
Example. f(x) = √(x − 2). The square root needs x − 2 ≥ 0, so the domain is [2, ∞) and the range is [0, ∞).
Example. f(x) = x² + 1. The domain is ℝ; since x² ≥ 0, the range is [1, ∞).
Standard real functions
Function | Rule | Domain | Range |
|---|---|---|---|
Identity | f(x) = x | ℝ | ℝ |
Constant | f(x) = c | ℝ | {c} |
Polynomial | f(x) = a₀ + a₁x + … + aₙxⁿ | ℝ | depends |
Rational | f(x) = p(x)/q(x) | ℝ except zeroes of q | depends |
Modulus | f(x) = x if x ≥ 0, and −x if x < 0 | ℝ | [0, ∞) |
Signum | f(x) = 1 if x > 0, 0 if x = 0, −1 if x < 0 | ℝ | {−1, 0, 1} |
Greatest integer | f(x) = [x], the greatest integer ≤ x | ℝ | ℤ (the integers) |
The graph of the identity function is a straight line through the origin at 45°. The modulus graph is V-shaped. The greatest integer function has a step graph: [2.7] = 2, [−1.3] = −2, [5] = 5.
Algebra of real functions
For f, g : X → ℝ:
(f + g)(x) = f(x) + g(x)
(f − g)(x) = f(x) − g(x)
(fg)(x) = f(x) g(x)
(f/g)(x) = f(x)/g(x), provided g(x) ≠ 0
(αf)(x) = α f(x) for a real number α
Example. f(x) = x² and g(x) = 2x + 1. (f + g)(x) = x² + 2x + 1 = (x + 1)², and (f/g)(x) = x²/(2x + 1), for x ≠ −½.
Summary
A × B is the set of ordered pairs; n(A × B) = n(A) × n(B).
A relation is any subset of A × B; a function gives each element of the domain exactly one image.
Range ⊆ codomain; find the domain by excluding values where the formula fails.
Know the modulus, signum and greatest integer functions and their graphs.