Set Theory and Set Operations
Complete guide to sets, unions, intersections, differences, and complements with examples
Introduction to Sets
A set is a well-defined collection of distinct objects with a common property. Sets are fundamental to mathematics and are used to describe collections such as the integers, the points on a line, or even abstract concepts.
A set is a collection of distinct objects called elements. A set is completely determined by its elements, regardless of the order or repetitions in which they are written.
Notation and Basic Concepts
Sets are typically denoted by capital letters and their elements are listed within curly braces:
If a set \(A\) contains the numbers 0, 1, 4, and 9, we write:
\(\displaystyle A = \{0, 1, 4, 9\}\)
Element Membership
We use special symbols to indicate whether an element belongs to a set or not:
- \(\displaystyle 4 \in A\) means "4 is an element of A" or "4 belongs to A"
- \(\displaystyle 5 \notin A\) means "5 is not an element of A" or "5 does not belong to A"
Examples of Set Notation
Set A: \(\displaystyle A = \{0, 1, 4, 9\}\)
- \(\displaystyle 1 \in A\) ✓ (1 is in the set)
- \(\displaystyle 4 \in A\) ✓ (4 is in the set)
- \(\displaystyle 5 \notin A\) ✓ (5 is not in the set)
- \(\displaystyle 9 \in A\) ✓ (9 is in the set)
Set Equality and Subsets
Set Equality
Two sets are equal if they contain exactly the same elements. The order of elements and repetitions do not matter.
Two sets \(A\) and \(B\) are equal (written \(\displaystyle A = B\)) if and only if they contain exactly the same elements.
Example: Set Equality
The following sets are equal:
Both sets contain the same elements: 0, 1, 4, and 9
Subsets
A subset is a set whose elements are all contained in another set.
A set \(B\) is a subset of a set \(A\) (written \(\displaystyle B \subseteq A\)) if every element of \(B\) is also an element of \(A\).
Example: Subsets
Given \(\displaystyle A = \{0, 1, 4, 9\}\):
- \(\displaystyle \{1, 4\} \subseteq A\) ✓ (both 1 and 4 are in A)
- \(\displaystyle \{0, 1, 4, 9\} \subseteq A\) ✓ (A is a subset of itself)
- \(\displaystyle \{1, 5\} \not\subseteq A\) (5 is not in A)
The Empty Set
The empty set is a special set that contains no elements. It is a subset of every set.
The empty set, denoted by \(\displaystyle \emptyset\) or \(\displaystyle \{\}\), is the set that contains no elements. For any set \(A\), we have \(\displaystyle \emptyset \subseteq A\).
Example of Empty Set
The set of all real numbers that satisfy \(\displaystyle x^2 + 1 = 0\) is the empty set, because no real number squared equals -1.
Union of Sets
The union of two sets is a new set that contains all elements that belong to either set (or both).
The union of sets \(A\) and \(B\), denoted \(\displaystyle A \cup B\), is the set of all elements that are in \(A\) or in \(B\) or in both:
\(\displaystyle A \cup B = \{x : x \in A \text{ or } x \in B\}\)
Example of Union
Given:
- \(\displaystyle A = \{0, 1, 4, 9\}\)
- \(\displaystyle B = \{2, 5, 9\}\)
The union is:
All elements from both sets, with 9 listed only once (sets contain distinct elements)
Intersection of Sets
The intersection of two sets is a new set that contains only the elements that belong to both sets.
The intersection of sets \(A\) and \(B\), denoted \(\displaystyle A \cap B\), is the set of all elements that are in both \(A\) and \(B\):
\(\displaystyle A \cap B = \{x : x \in A \text{ and } x \in B\}\)
Example of Intersection
Given:
- \(\displaystyle A = \{0, 1, 4, 9\}\)
- \(\displaystyle B = \{2, 5, 9\}\)
The intersection is:
Only 9 is in both sets
Two sets are called disjoint if their intersection is empty: \(\displaystyle A \cap B = \emptyset\). This means they have no elements in common.
Difference of Sets
The difference of two sets is a new set that contains elements from the first set that are not in the second set.
The difference of sets \(A\) and \(B\), denoted \(\displaystyle A \setminus B\) or \(\displaystyle A - B\), is the set of all elements that are in \(A\) but not in \(B\):
\(\displaystyle A \setminus B = \{x : x \in A \text{ and } x \notin B\}\)
Example of Difference
Given:
- \(\displaystyle A = \{0, 1, 4, 9\}\)
- \(\displaystyle B = \{2, 5, 9\}\)
The difference is:
Elements in A that are not in B: 0, 1, and 4 are in A but not in B
Complement of a Set
The complement of a set is the set of all elements in a universal set that are not in the given set.
Given a universal set \(\displaystyle U\) and a subset \(\displaystyle A \subseteq U\), the complement of \(A\), denoted \(\displaystyle A^c\) or \(\displaystyle \overline{A}\), is the set of all elements in \(U\) that are not in \(A\):
\(\displaystyle A^c = \{x : x \in U \text{ and } x \notin A\}\)
or equivalently \(\displaystyle A^c = U \setminus A\)
Example of Complement
Given:
- Universal set: \(\displaystyle U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)
- Subset: \(\displaystyle A = \{0, 1, 4, 9\}\)
The complement is:
All elements in U that are not in A
Summary of Set Operations
| Operation | Notation | Definition | Example |
|---|---|---|---|
| Union | \(\displaystyle A \cup B\) | Elements in A or B or both | \(\displaystyle \{1,2\} \cup \{2,3\} \) \(\displaystyle = \{1,2,3\}\) |
| Intersection | \(\displaystyle A \cap B\) | Elements in both A and B | \(\displaystyle \{1,2\} \cap \{2,3\}\)\(\displaystyle = \{2\}\) |
| Difference | \(\displaystyle A \setminus B\) | Elements in A but not in B | \(\displaystyle \{1,2\} \setminus \{2,3\}\)\(\displaystyle = \{1\}\) |
| Complement | \(\displaystyle A^c\) | Elements in U but not in A | \(\displaystyle \text{If } U = \{1,2,3\},\)\(\displaystyle A = \{1\}, \)\(\displaystyle \text{ then } A^c = \{2,3\}\) |
| Subset | \(\displaystyle A \subseteq B\) | All elements of A are in B | \(\displaystyle \{1,2\} \subseteq \{1,2,3\}\) |
| Empty Set | \(\displaystyle \emptyset\) | Set with no elements | \(\displaystyle \{1,2\} \cap \{3,4\}\)\(\displaystyle = \emptyset\) |
Properties of Set Operations
Commutative Laws
Associative Laws
and \(\displaystyle (A \cap B) \cap C = A \cap (B \cap C)\)
Distributive Laws
and \(\displaystyle A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)
De Morgan's Laws
and \(\displaystyle (A \cap B)^c = A^c \cup B^c\)
Venn Diagrams
Venn diagrams are a visual way to represent sets and their operations. Sets are represented as circles or other shapes within a rectangle representing the universal set.
In a Venn diagram:
- The rectangle represents the universal set \(U\)
- Each set is represented by a circle or region
- Overlapping regions show elements in the intersection
- Non-overlapping regions show elements unique to each set
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