Formula sheet: Relations and Functions
Relations and Functions16 min readLesson 1 of 6
1.1 Ordered pairs and products
An ordered pair groups two objects in a fixed order. Two ordered pairs are equal only when both positions match, so (a, b) = (c, d) forces a = c and b = d. The Cartesian product of two sets collects every ordered pair with first entry from P and second from Q. If either set is empty, the product is empty as well.
| Rule | Statement |
|---|---|
| Product | P x Q = pairs (p, q) with p in P and q in Q |
| Size of product | n(P x Q) = n(P) times n(Q) |
| Empty case | A x phi = phi |
| Order matters | A x B is not equal to B x A in general |
| Plane | R x R gives the coordinates of every point in two dimensional space |
| Space | R x R x R gives triples of coordinates in three dimensional space |
1.2 Relations
A relation from A to B is any subset of the product A x B. A rule picks out some pairs; the rest are left out. For a relation R, the set of first entries is the domain, the set of second entries is the range, and the whole target set B is the codomain. The range always sits inside the codomain.
A relation from A to A is called a relation on A. An arrow diagram draws the two copies of the set and joins each related pair with an arrow. When the first set has p members and the second has q, the product holds pq pairs. Each subset of those pairs is a relation, so the count of relations is 2 raised to the power pq.
1.3 The function test
A relation f from A to B is a function when every element of A has exactly one image in B. Two failures are possible: some element of A has no image at all, or one element of A points to two different images. Either failure disqualifies the relation.
For a function f: A to B, the domain of f is the whole of A, the codomain is B and the range is the set of images actually produced. A test question gives a table, a graph or a roster of pairs; scan the first coordinates and look for a missing element or a repeated first coordinate with a different partner.
A real valued function has its range inside the real numbers. A real function has both its domain and its range inside the real numbers.
1.4 Standard real functions and graphs
| Function | Rule | Domain | Range |
|---|---|---|---|
| Identity | f(x) = x | R | R |
| Constant | f(x) = c | R | the single value c |
| Linear | f(x) = mx + c | R | R when m is not zero |
| Square | f(x) = x squared | R | all values at least zero |
| Cube | f(x) = x cubed | R | R |
| Reciprocal | f(x) = 1/x | R without 0 | R without 0 |
| Modulus | f(x) = | x | |
| Signum | f(x) = sign of x | R | minus 1, 0, 1 |
| Greatest integer | f(x) = bracket x | R | the integers |
The graphs of these functions are asked often. The quadratic graph is a U shaped curve with its lowest point at the origin, while the cube curve passes through the origin and rises on both ends. The modulus graph is V shaped, and the greatest integer graph is a step function.
1.5 Algebra of real functions
Suppose f and g are real functions with a common domain X. The five combinations below work point by point on that domain.
| Combination | Rule | Extra condition |
|---|---|---|
| Sum | (f + g)(x) = f(x) + g(x) | none |
| Difference | (f - g)(x) = f(x) - g(x) | none |
| Scalar multiple | (k f)(x) = k f(x) | k is a real number |
| Product | (f g)(x) = f(x) g(x) | none |
| Quotient | (f / g)(x) = f(x) / g(x) | g(x) is not zero |
Only the quotient removes points from the domain: solve g(x) = 0 and exclude those values.
Exam
For a domain question with a fraction, solve the denominator equal to zero first and exclude those values. For a square root, put the expression inside the root as at least zero. These two checks cover most domain questions on this chapter.