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 \}
Length of a set
- The length of a set is defined as the number of elements it contains.
- It is denoted by surrounding the set by a pair of vertical lines:
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 (Python)
A = frozenset([1, 2, 3, 4, 5])
print(1 in A) # prints 'True'
print(6 not in A) # prints 'True'