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Important Sets
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Sets
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Introduction
  • A set is a collection of elements, such as:
    • A = {1, 2, 3, 4, 5}
  • A set does not have order:
    • {1, 2, 3, 4, 5} is the same as {5, 4, 3, 2, 1}
    • It is convenient to write the elements of a set in consecutive order, but this is only for convenience.
  • Every element in a set is unique, multiples of the same element are ignored:
    • The set {1, 2, 2} has a size of 2, and is the same as {1, 2}
Notation
  • It is common for sets to be denoted by a uppercase letter, and for its elements to be wrapped in curly braces:k

    LaTeX: \mathbf{A} = \{ 1, 2, 3, 4, 5 \}
  • Sets can also be defined using set builder notation:
    • A rule is used to show which elements are members of the set:
    • The set A defined above would contain all even integers between 0 and 2000.
    • The format of the rule is: (formula: conditions) or (formula| conditions).
    • If it is not specified as a condition, then it is assumed that a is a real number.
Elements in a set
  • Elements in a set are usually represented by a lowercase letter.
  • To demonstrate that an element is part of a set, we use the set membership symbol:

    LaTeX: \mathbf{a} \in \mathbf{A}
  • This can also be done with proper elements:

    LaTeX: \1 \in \{ 1, 2, 3, 4, 5 \}
  • To show that an element is not part of a set, we use the 'not member of' symbol:

    LaTeX: 6 \notin \{ 1, 2, 3, 4, 5 \}
The Empty Set

The Empty Set (or Null Set) is a set containing no items. It is represented by the empty set symbol:


LaTeX: \varnothing = \{\}
Code
A = frozenset([1, 2, 3, 4, 5])

print(1 in A) # prints 'True'
print(6 not in A) # prints 'True'

Important Sets

Introduction
  • Some sets are used so often in mathematics that they are given their own names.
  • This article will cover two of them:
    • Z: The set of all integers.
    • R: The set of all real numbers.
Integers
  • The set of all integers contains all the whole numbers, including negative numbers and 0.
  • It is represented by an uppercase bold (or blackboard bold) Z:

    LaTeX: \mathbb{Z} = \{ ..., -3, -2, -1, 0, 1, 2, 3, ... \}
Code
i = 1
print(type(i))  # prints <class 'int'>
Real Numbers
  • The set of all real numbers contains every number that is not imaginary. This includes:
    • Integers: e.g. 0, 1, 2
    • Rational Numbers: Those which can be represented as a fraction, e.g. 0.5, 1.1
    • Irrational Numbers: Those which cannot be represented by a fraction, e.g.
  • It is represented by an uppercase bold (or blackboard bold) R:

    LaTeX: \mathbb{R} = \{ -1, 1.4, \pi \}
Code
Unfortunately we cannot represent irrational numbers using code. Instead we can approximate them as floating point numbers:
r = 1.41421356237
print(type(r))  # prints <class 'float'>
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