Definition of Sets
A set is a collection of well-defined, distinct objects called elements or members. Sets are typically denoted by capital letters with elements enclosed in curly braces.
Given set A containing odd numbers less than 10:
\[ A = \{1, 3, 5, 7, 9\} \]Membership Notation:
For any element x:
- If \( x \) is in set \( A \): \( x \in A \)
- If \( x \) is not in set \( A \): \( x \notin A \)
Well-Defined:
A set must have unambiguous membership criteria. For example, "books in this room" is well-defined, while "interesting books" is not.
Complement of a Set (\( A' \))
The complement of set \( A \) contains all elements not in \( A \) within a given universal set.
If \( \mathbb{U} = \{1, 2, 3, \ldots, 10\} \) and \( A = \{1, 3, 5, 7, 9\} \), then:
\[ A' = \{2, 4, 6, 8, 10\} \]Universal Set (\( \mathbb{U} \))
The set containing all elements under consideration. It satisfies:
\[ A \cup A' = \mathbb{U} \]Cardinality (\( n(A) \))
The number of elements in a set:
\[ n(A) = 5 \quad \text{(for our example set)} \]Finite Set:
Has countable elements (e.g., \( \{1, 2, 3\} \))
Infinite Set:
Has uncountable elements (e.g., \( \mathbb{N} = \{1, 2, 3, \ldots\} \))
\[ n(\mathbb{N}) = \infty \]Describing Sets
1. Word Description
Using natural language to define the set:
2. Listing (Roster)
Explicitly listing all elements:
3. Set-Builder Notation
Mathematical rule definition:
Read as: "x such that x is between 20 and 30 inclusive, where x is a whole number"
Special Sets
Empty Set (\( \emptyset \))
Contains no elements:
\[ \emptyset = \{\} \] \[ n(\emptyset) = 0 \]Unit Set
Contains exactly one element:
\[ S = \{a\} \]Practice Exercise
Question 1
Given \( \mathbb{U} = \{1, 2, \ldots, 10\} \) and \( A = \{2, 4, 6, 8, 10\} \), find \( A' \).
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