Sets and Set Notation

Overview

Recall questions on what a set is, the two notations for specifying one, and the meaning of the symbols in a set comprehension.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What is a set?

   Model answer

   A set is an unordered collection in which no value appears more than once. Because it is unordered, {1, 2, 3} and {3, 1, 2} are the same set, and because each value occurs at most once, writing a value twice adds nothing.

2. 2

   Give the two notations for specifying a set, with an example of each.

   Model answer

   Either list the members between curly braces — A = {1, 2, 3, 4, 5} — or define them by a property they satisfy, which is set comprehension: A = {x | x ∈ ℕ ∧ x ≥ 1}. The second form says that A contains exactly those x for which the stated condition holds, here every natural number of at least 1. Listing suits small finite sets; comprehension is the only workable option for very large or infinite ones.

3. 3

   In the set comprehension A = {x | x ∈ ℕ ∧ x ≥ 1}, what does each symbol mean?

   Model answer

   The vertical bar | is read 'such that'. The symbol ∈ is the membership test, so x ∈ ℕ asserts that x belongs to ℕ. ℕ names the natural numbers, {0, 1, 2, 3, 4, …}. The symbol ∧ is logical AND, joining the two conditions, so x ≥ 1 imposes a second requirement: x must be at least 1. Taken as a whole, the expression denotes every x that is both a natural number and no smaller than 1 — that is, {1, 2, 3, 4, …}, with 0 excluded by the second condition.

4. 4

   Write in set comprehension notation the set of even natural numbers greater than 10, and list its first four elements.

   Model answer

   E = {x | x ∈ ℕ ∧ x > 10 ∧ x mod 2 = 0}. Read: the set of all x such that x is a natural number and x is greater than 10 and x divides exactly by 2. Its first four elements are 12, 14, 16 and 18.

5. 5

   How does Python let you construct a set, and what is a set comprehension in Python?

   Model answer

   Curly braces build a set object directly, so {1, 2, 3} is a set of three elements. A set comprehension derives a new set from an existing one by stating a rule for its members: evaluating {2 * x for x in {1, 2, 3}} yields {2, 4, 6}, each element being double the corresponding element of the original. Python describes this as a comprehension over {1, 2, 3}, and it is the programming counterpart of the mathematician's {2x | x ∈ {1, 2, 3}}.

6. 6

   What is the empty set, and what are the two ways of writing it?

   Model answer

   The empty set is the set with no elements. It is written {} or, using the alternative symbol, Ø.

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