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:
\[ S = \{a\} \]Types of Sets
Equal Sets
Two sets are equal if they contain exactly the same elements, regardless of order.
Example:
\[ A = \{1, 2, 3\}, \quad B = \{3, 2, 1\} \] \[ A = B \]Subsets
Set \( A \) is a subset of \( B \) if all elements of \( A \) are in \( B \):
\[ A \subseteq B \]Example:
\[ A = \{1, 2\}, \quad B = \{1, 2, 3, 4\} \] \[ A \subseteq B \]Proper Subset:
When \( A \subseteq B \) but \( A \neq B \):
\[ A \subset B \]Disjoint Sets
Sets with no common elements:
\[ A \cap B = \emptyset \]Example:
\[ A = \{1, 3, 5\}, \quad B = \{2, 4, 6\} \] \[ A \cap B = \emptyset \]Number Systems
Standard sets of numbers used in mathematics:
Natural Numbers (\( \mathbb{N} \))
\[ \mathbb{N} = \{1, 2, 3, \ldots\} \]Whole Numbers (\( \mathbb{W} \))
\[ \mathbb{W} = \{0, 1, 2, 3, \ldots\} \]Integers (\( \mathbb{Z} \))
\[ \mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\} \]Rational Numbers (\( \mathbb{Q} \))
\[ \mathbb{Q} = \left\{\frac{p}{q} \mid p,q \in \mathbb{Z}, q \neq 0\right\} \]Real Numbers (\( \mathbb{R} \))
All numbers on the number line
Relationships between number systems:
\[ \mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \]Set Operations
Union (\( A \cup B \))
Combines all elements from both sets:
\[ A \cup B = \{x \mid x \in A \text{ or } x \in B\} \]Example:
\[ A = \{1, 2, 3\}, \quad B = \{3, 4, 5\} \] \[ A \cup B = \{1, 2, 3, 4, 5\} \]Intersection (\( A \cap B \))
Contains only elements common to both sets:
\[ A \cap B = \{x \mid x \in A \text{ and } x \in B\} \]Example:
\[ A = \{1, 2, 3\}, \quad B = \{3, 4, 5\} \] \[ A \cap B = \{3\} \]Difference (\( A \setminus B \))
Elements in \( A \) but not in \( B \):
\[ A \setminus B = \{x \mid x \in A \text{ and } x \notin B\} \]Example:
\[ A = \{1, 2, 3\}, \quad B = \{3, 4, 5\} \] \[ A \setminus B = \{1, 2\} \]Symmetric Difference (\( A \Delta B \))
Elements in either set but not in both:
\[ A \Delta B = (A \setminus B) \cup (B \setminus A) \]Example:
\[ A = \{1, 2, 3\}, \quad B = \{3, 4, 5\} \] \[ A \Delta B = \{1, 2, 4, 5\} \]Venn Diagrams
Visual representation of set relationships:
Union:
All shaded areas
Intersection:
Overlapping area
Complement:
Area outside the set
Three-Set Problems
Problems involving three sets with overlapping regions:
Cardinality formula for three sets:
\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \]Example Problem:
In a class of 50 students:
- 30 study Math
- 25 study Physics
- 20 study Chemistry
- 10 study Math and Physics
- 8 study Math and Chemistry
- 7 study Physics and Chemistry
- 5 study all three subjects
How many students study none of these subjects?
Using the formula:
\[ n(M \cup P \cup C) = 30 + 25 + 20 - 10 - 8 - 7 + 5 = 55 \]Total students = 50
\[ \text{Students studying none} = 50 - 55 + 5 = 0 \]Practice Exercise
Question 1
Given \( \mathbb{U} = \{1, 2, \ldots, 10\} \) and \( A = \{2, 4, 6, 8, 10\} \), find \( A' \).
Question 2
If \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), find \( A \Delta B \).
Question 3
For sets \( X = \{a, b, c\} \) and \( Y = \{c, d, e\} \), verify that \( n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \).
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