Formula sheet: Sets
Sets16 min readLesson 1 of 7
1.1 Notation you must read correctly
Any well-defined collection of objects counts as a set. Sets get capital letters and their members get lower case letters. The symbol in means belongs to, and the crossed version means does not belong. Roster form lists the elements inside braces, separated by commas. Set-builder form gives a rule that every member satisfies and no outsider satisfies.
| Symbol | Read as |
|---|---|
| x in A | x is an element of A |
| A is a subset of B | every element of A lies in B |
| A is a proper subset of B | A lies inside B and A is not equal to B |
| A union B | elements in A, or in B, or in both |
| A intersection B | elements common to A and B |
| A minus B | elements in A that are not in B |
| A prime | elements of U that are not in A |
| n(A) | number of distinct elements in A |
1.2 Standard number sets
The five family sets below appear in almost every question. Learn both the symbol and the description.
| Set | Members |
|---|---|
| N | natural numbers 1, 2, 3, and so on |
| Z | integers, including 0 and the negatives |
| Q | rational numbers p/q with q not zero |
| R | all real numbers, rational and irrational |
| T | irrational numbers, the real numbers that are not rational |
Their nesting is part of the syllabus: N is a subset of Z, Z is a subset of Q, and Q is a subset of R. Also T is a subset of R, while no natural number is irrational.
1.3 Interval notation
Intervals are the subsets of R that appear throughout calculus and coordinate geometry. The bracket tells you whether an endpoint is included.
| Interval | Set-builder form | Endpoints |
|---|---|---|
| (a, b) | a < x < b | both excluded |
| [a, b] | a <= x <= b | both included |
| [a, b) | a <= x < b | a included, b excluded |
| (a, b] | a < x <= b | a excluded, b included |
| [0, infinity) | x >= 0 | non-negative reals |
| (minus infinity, 0) | x < 0 | negative reals |
The length of any of these four bounded intervals is b - a. Infinity is never closed with a square bracket because it is not a number you can reach.
1.4 Operations and their laws
The four operations take two sets and return a new set. The laws listed below let you simplify expressions without listing elements.
| Law | Union form | Intersection form |
|---|---|---|
| Commutative | A union B = B union A | A intersection B = B intersection A |
| Associative | (A union B) union C = A union (B union C) | (A intersection B) intersection C = A intersection (B intersection C) |
| Identity | A union phi = A | A intersection U = A |
| Idempotent | A union A = A | A intersection A = A |
| Absorption | A union (A intersection B) = A | A intersection (A union B) = A |
Two more rules carry most of the algebra in board questions. Intersection distributes over union and union distributes over intersection. De Morgan's laws swap the operation when a complement crosses a bracket.
1.5 Counting rules that follow from the definitions
The counting formulas are not extra material to memorise separately; each one follows from the pair of sets being disjoint after you split them. For two sets, subtract the double counted overlap. For three sets, add the three single counts, subtract the three pairwise overlaps, and add back the triple overlap. A set with n elements has 2 to the power n subsets, and the empty set has exactly one subset, itself.
If A and B are disjoint, the overlap term drops out and n(A union B) is simply n(A) + n(B). For three pairwise disjoint sets the same gives the plain sum.
Exam
In a one mark question, a correct answer with no working still needs the right interval type or the right bracket. Write the interval or the set in roster form exactly as asked, because the format is part of the answer.