Sets - Ninth Grade Math
Introduction to Sets
Set: A well-defined collection of distinct objects, numbers, or elements
Element/Member: An object in a set (denoted by $\in$)
Universal Set (U): The set containing all elements under consideration
Empty/Null Set (∅ or { }): A set with no elements
Cardinality: The number of elements in a set, denoted as $|A|$ or $n(A)$
Element/Member: An object in a set (denoted by $\in$)
Universal Set (U): The set containing all elements under consideration
Empty/Null Set (∅ or { }): A set with no elements
Cardinality: The number of elements in a set, denoted as $|A|$ or $n(A)$
Ways to Represent Sets:
1. Roster/Tabular Form: List all elements in braces
Example: $A = \{1, 2, 3, 4, 5\}$
2. Set-Builder Notation: Describe properties of elements
Example: $A = \{x | x \text{ is a natural number less than 6}\}$
3. Venn Diagram: Visual representation using circles
1. Roster/Tabular Form: List all elements in braces
Example: $A = \{1, 2, 3, 4, 5\}$
2. Set-Builder Notation: Describe properties of elements
Example: $A = \{x | x \text{ is a natural number less than 6}\}$
3. Venn Diagram: Visual representation using circles
| Notation | Symbol | Meaning | Example |
|---|---|---|---|
| Element of | $\in$ | $x$ is an element of set $A$ | $3 \in \{1, 2, 3, 4\}$ |
| Not an element of | $\notin$ | $x$ is not an element of set $A$ | $5 \notin \{1, 2, 3, 4\}$ |
| Empty set | $\emptyset$ or $\{\}$ | Set with no elements | $A = \emptyset$ |
| Cardinality | $|A|$ or $n(A)$ | Number of elements in $A$ | $|\{1, 2, 3\}| = 3$ |
1. Sets, Subsets, and Complements
Subsets
Subset: A set $A$ is a subset of set $B$ if every element of $A$ is also in $B$
Notation: $A \subseteq B$ (read as "A is a subset of B")
Proper Subset: $A$ is a proper subset of $B$ if $A \subseteq B$ and $A \neq B$
Notation: $A \subset B$ (A is a proper subset of B)
Notation: $A \subseteq B$ (read as "A is a subset of B")
Proper Subset: $A$ is a proper subset of $B$ if $A \subseteq B$ and $A \neq B$
Notation: $A \subset B$ (A is a proper subset of B)
Subset Definition:
$$A \subseteq B \text{ if and only if } x \in A \Rightarrow x \in B$$
(If $x$ is in $A$, then $x$ must be in $B$)
$$A \subseteq B \text{ if and only if } x \in A \Rightarrow x \in B$$
(If $x$ is in $A$, then $x$ must be in $B$)
Important Subset Properties:
• Every set is a subset of itself: $A \subseteq A$
• The empty set is a subset of every set: $\emptyset \subseteq A$
• If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$ (Transitive)
• If $A \subseteq B$ and $B \subseteq A$, then $A = B$
• Number of subsets of a set with $n$ elements = $2^n$
• Every set is a subset of itself: $A \subseteq A$
• The empty set is a subset of every set: $\emptyset \subseteq A$
• If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$ (Transitive)
• If $A \subseteq B$ and $B \subseteq A$, then $A = B$
• Number of subsets of a set with $n$ elements = $2^n$
Example 1: Let $A = \{1, 2\}$ and $B = \{1, 2, 3, 4\}$
Is $A \subseteq B$?
Yes, because every element of $A$ (1 and 2) is also in $B$.
We write: $A \subseteq B$ or $A \subset B$ (proper subset)
Is $A \subseteq B$?
Yes, because every element of $A$ (1 and 2) is also in $B$.
We write: $A \subseteq B$ or $A \subset B$ (proper subset)
Example 2: List all subsets of $A = \{a, b\}$
Subsets:
• $\emptyset$ (empty set)
• $\{a\}$
• $\{b\}$
• $\{a, b\}$ (the set itself)
Total subsets: $2^2 = 4$ subsets
Subsets:
• $\emptyset$ (empty set)
• $\{a\}$
• $\{b\}$
• $\{a, b\}$ (the set itself)
Total subsets: $2^2 = 4$ subsets
Example 3: How many subsets does $\{1, 2, 3\}$ have?
Formula: Number of subsets = $2^n$ where $n$ = number of elements
$n = 3$, so number of subsets = $2^3 = 8$
They are: $\emptyset$, $\{1\}$, $\{2\}$, $\{3\}$, $\{1, 2\}$, $\{1, 3\}$, $\{2, 3\}$, $\{1, 2, 3\}$
Formula: Number of subsets = $2^n$ where $n$ = number of elements
$n = 3$, so number of subsets = $2^3 = 8$
They are: $\emptyset$, $\{1\}$, $\{2\}$, $\{3\}$, $\{1, 2\}$, $\{1, 3\}$, $\{2, 3\}$, $\{1, 2, 3\}$
Complement of a Set
Complement: The complement of set $A$ (denoted as $A'$ or $A^c$) contains all elements in the universal set $U$ that are NOT in $A$
Complement Notation:
$$A' = \{x \in U : x \notin A\}$$
or
$$A' = U - A$$
Read as: "A prime equals all elements in U that are not in A"
$$A' = \{x \in U : x \notin A\}$$
or
$$A' = U - A$$
Read as: "A prime equals all elements in U that are not in A"
Properties of Complement:
• $(A')' = A$ (Complement of complement is the original set)
• $A \cup A' = U$ (A union with its complement gives universal set)
• $A \cap A' = \emptyset$ (A intersection with its complement is empty)
• $U' = \emptyset$ (Complement of universal set is empty set)
• $\emptyset' = U$ (Complement of empty set is universal set)
• If $A \subseteq B$, then $B' \subseteq A'$
• $(A')' = A$ (Complement of complement is the original set)
• $A \cup A' = U$ (A union with its complement gives universal set)
• $A \cap A' = \emptyset$ (A intersection with its complement is empty)
• $U' = \emptyset$ (Complement of universal set is empty set)
• $\emptyset' = U$ (Complement of empty set is universal set)
• If $A \subseteq B$, then $B' \subseteq A'$
Example 1: Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $A = \{2, 4, 6, 8, 10\}$ (even numbers)
Find $A'$:
$A' = \{1, 3, 5, 7, 9\}$ (all odd numbers in U)
Find $A'$:
$A' = \{1, 3, 5, 7, 9\}$ (all odd numbers in U)
Example 2: Let $U = \{x | x \text{ is a natural number} \leq 20\}$ and $A = \{2, 3, 5, 7, 11, 13, 17, 19\}$ (prime numbers)
Find $A'$:
$A' = \{1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20\}$ (non-prime numbers)
Find $A'$:
$A' = \{1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20\}$ (non-prime numbers)
Example 3: If $U = \{a, b, c, d, e, f\}$ and $B = \{a, c, e\}$, find $(B')'$
Step 1: Find $B'$
$B' = \{b, d, f\}$
Step 2: Find $(B')'$
$(B')' = \{a, c, e\} = B$
Conclusion: The complement of the complement is the original set!
Step 1: Find $B'$
$B' = \{b, d, f\}$
Step 2: Find $(B')'$
$(B')' = \{a, c, e\} = B$
Conclusion: The complement of the complement is the original set!
2. Set-Builder Notation
Set-Builder Notation: A mathematical notation for describing a set by stating the properties that its elements must satisfy
General Form: $\{x | \text{condition}\}$ or $\{x : \text{condition}\}$
Read as: "The set of all $x$ such that [condition]"
General Form: $\{x | \text{condition}\}$ or $\{x : \text{condition}\}$
Read as: "The set of all $x$ such that [condition]"
General Format:
$$\{x \in D | P(x)\}$$
where:
• $x$ = variable representing elements
• $D$ = domain (set from which elements come)
• $|$ or $:$ = "such that"
• $P(x)$ = property/condition that $x$ must satisfy
$$\{x \in D | P(x)\}$$
where:
• $x$ = variable representing elements
• $D$ = domain (set from which elements come)
• $|$ or $:$ = "such that"
• $P(x)$ = property/condition that $x$ must satisfy
Common Notations in Set-Builder Form:
• $\mathbb{N}$ = Natural numbers $\{1, 2, 3, 4, ...\}$
• $\mathbb{W}$ = Whole numbers $\{0, 1, 2, 3, ...\}$
• $\mathbb{Z}$ = Integers $\{..., -2, -1, 0, 1, 2, ...\}$
• $\mathbb{Q}$ = Rational numbers
• $\mathbb{R}$ = Real numbers
• $\mathbb{N}$ = Natural numbers $\{1, 2, 3, 4, ...\}$
• $\mathbb{W}$ = Whole numbers $\{0, 1, 2, 3, ...\}$
• $\mathbb{Z}$ = Integers $\{..., -2, -1, 0, 1, 2, ...\}$
• $\mathbb{Q}$ = Rational numbers
• $\mathbb{R}$ = Real numbers
| Roster Form | Set-Builder Notation | Description |
|---|---|---|
| $\{1, 2, 3, 4, 5\}$ | $\{x | x \in \mathbb{N}, x \leq 5\}$ | Natural numbers up to 5 |
| $\{2, 4, 6, 8, 10\}$ | $\{x | x = 2n, n \in \mathbb{N}, n \leq 5\}$ | Even numbers up to 10 |
| $\{1, 3, 5, 7, 9\}$ | $\{x | x = 2n-1, n \in \mathbb{N}, n \leq 5\}$ | Odd numbers up to 9 |
| $\{1, 4, 9, 16, 25\}$ | $\{x | x = n^2, n \in \mathbb{N}, n \leq 5\}$ | Perfect squares up to 25 |
Example 1: Write in roster form: $\{x | x \in \mathbb{Z}, -3 < x < 4\}$
Solution:
This represents all integers between -3 and 4 (not including -3 and 4)
Roster form: $\{-2, -1, 0, 1, 2, 3\}$
Solution:
This represents all integers between -3 and 4 (not including -3 and 4)
Roster form: $\{-2, -1, 0, 1, 2, 3\}$
Example 2: Write in set-builder notation: $\{10, 15, 20, 25, 30\}$
Solution:
These are multiples of 5 between 10 and 30
Set-builder: $\{x | x = 5n, n \in \mathbb{N}, 2 \leq n \leq 6\}$
or $\{x | x \text{ is a multiple of 5}, 10 \leq x \leq 30\}$
Solution:
These are multiples of 5 between 10 and 30
Set-builder: $\{x | x = 5n, n \in \mathbb{N}, 2 \leq n \leq 6\}$
or $\{x | x \text{ is a multiple of 5}, 10 \leq x \leq 30\}$
Example 3: Write in set-builder notation: $\{2, 3, 5, 7, 11\}$
Solution:
These are prime numbers less than 12
Set-builder: $\{x | x \text{ is a prime number}, x < 12\}$
Solution:
These are prime numbers less than 12
Set-builder: $\{x | x \text{ is a prime number}, x < 12\}$
Example 4: Express in roster form: $\{x | x^2 = 25\}$
Solution:
We need to find all values where $x^2 = 25$
$x = 5$ or $x = -5$
Roster form: $\{-5, 5\}$
Solution:
We need to find all values where $x^2 = 25$
$x = 5$ or $x = -5$
Roster form: $\{-5, 5\}$
Tips for Set-Builder Notation:
• Use $|$ or $:$ to mean "such that"
• Specify the domain (e.g., $x \in \mathbb{R}$, $x \in \mathbb{N}$)
• State the condition clearly
• Can use inequalities, equations, or descriptions
• Use $|$ or $:$ to mean "such that"
• Specify the domain (e.g., $x \in \mathbb{R}$, $x \in \mathbb{N}$)
• State the condition clearly
• Can use inequalities, equations, or descriptions
3. Interval Notation
Interval Notation: A way to represent a set of real numbers between two endpoints
Brackets [ ]: Square brackets mean the endpoint IS included (closed)
Parentheses ( ): Round parentheses mean the endpoint is NOT included (open)
Infinity (∞): Always uses parentheses, never brackets
Brackets [ ]: Square brackets mean the endpoint IS included (closed)
Parentheses ( ): Round parentheses mean the endpoint is NOT included (open)
Infinity (∞): Always uses parentheses, never brackets
Interval Notation Types:
1. Closed Interval: $[a, b]$
Includes both endpoints
Set-builder: $\{x | a \leq x \leq b\}$
2. Open Interval: $(a, b)$
Excludes both endpoints
Set-builder: $\{x | a < x < b\}$
3. Half-Open Interval: $[a, b)$ or $(a, b]$
Includes one endpoint, excludes the other
Set-builder: $\{x | a \leq x < b\}$ or $\{x | a < x \leq b\}$
4. Infinite Intervals:
$(a, \infty)$ means $\{x | x > a\}$
$[a, \infty)$ means $\{x | x \geq a\}$
$(-\infty, b)$ means $\{x | x < b\}$
$(-\infty, b]$ means $\{x | x \leq b\}$
$(-\infty, \infty)$ means all real numbers $\mathbb{R}$
1. Closed Interval: $[a, b]$
Includes both endpoints
Set-builder: $\{x | a \leq x \leq b\}$
2. Open Interval: $(a, b)$
Excludes both endpoints
Set-builder: $\{x | a < x < b\}$
3. Half-Open Interval: $[a, b)$ or $(a, b]$
Includes one endpoint, excludes the other
Set-builder: $\{x | a \leq x < b\}$ or $\{x | a < x \leq b\}$
4. Infinite Intervals:
$(a, \infty)$ means $\{x | x > a\}$
$[a, \infty)$ means $\{x | x \geq a\}$
$(-\infty, b)$ means $\{x | x < b\}$
$(-\infty, b]$ means $\{x | x \leq b\}$
$(-\infty, \infty)$ means all real numbers $\mathbb{R}$
| Interval Notation | Inequality | Set-Builder Notation | Number Line |
|---|---|---|---|
| $[2, 5]$ | $2 \leq x \leq 5$ | $\{x | 2 \leq x \leq 5\}$ | Closed circles at 2 and 5, shaded between |
| $(2, 5)$ | $2 < x < 5$ | $\{x | 2 < x < 5\}$ | Open circles at 2 and 5, shaded between |
| $[2, 5)$ | $2 \leq x < 5$ | $\{x | 2 \leq x < 5\}$ | Closed at 2, open at 5, shaded between |
| $(2, \infty)$ | $x > 2$ | $\{x | x > 2\}$ | Open circle at 2, shaded right |
| $(-\infty, 5]$ | $x \leq 5$ | $\{x | x \leq 5\}$ | Closed circle at 5, shaded left |
| $(-\infty, \infty)$ | All real numbers | $\{x | x \in \mathbb{R}\}$ | Entire number line shaded |
Example 1: Write in interval notation: $-3 \leq x < 7$
Solution:
• $x = -3$ is included → use bracket $[$
• $x = 7$ is not included → use parenthesis $)$
Answer: $[-3, 7)$
Solution:
• $x = -3$ is included → use bracket $[$
• $x = 7$ is not included → use parenthesis $)$
Answer: $[-3, 7)$
Example 2: Write the inequality for: $(4, 10]$
Solution:
• Parenthesis at 4 → 4 not included
• Bracket at 10 → 10 is included
Answer: $4 < x \leq 10$
Solution:
• Parenthesis at 4 → 4 not included
• Bracket at 10 → 10 is included
Answer: $4 < x \leq 10$
Example 3: Write in interval notation: $x \geq -5$
Solution:
• $x = -5$ is included → use bracket $[$
• No upper limit → use $\infty$ with parenthesis
Answer: $[-5, \infty)$
Solution:
• $x = -5$ is included → use bracket $[$
• No upper limit → use $\infty$ with parenthesis
Answer: $[-5, \infty)$
Example 4: Write in set-builder notation: $(-2, 8]$
Solution:
Answer: $\{x | -2 < x \leq 8\}$ or $\{x \in \mathbb{R} | -2 < x \leq 8\}$
Solution:
Answer: $\{x | -2 < x \leq 8\}$ or $\{x \in \mathbb{R} | -2 < x \leq 8\}$
Key Points for Interval Notation:
• [ ] means "included" (≤ or ≥)
• ( ) means "not included" (< or >)
• Always use ( ) with ∞ or -∞
• Write smaller number first: [a, b] where a < b
• For all real numbers, write $(-\infty, \infty)$
• [ ] means "included" (≤ or ≥)
• ( ) means "not included" (< or >)
• Always use ( ) with ∞ or -∞
• Write smaller number first: [a, b] where a < b
• For all real numbers, write $(-\infty, \infty)$
4. Unions and Intersections of Sets
Union of Sets
Union: The union of two sets $A$ and $B$ (denoted as $A \cup B$) is the set of all elements that are in $A$, in $B$, or in both
Symbol: $\cup$ (cup symbol)
Keyword: "OR"
Symbol: $\cup$ (cup symbol)
Keyword: "OR"
Union Definition:
$$A \cup B = \{x | x \in A \text{ OR } x \in B\}$$
Read as: "A union B equals the set of all x such that x is in A or x is in B"
$$A \cup B = \{x | x \in A \text{ OR } x \in B\}$$
Read as: "A union B equals the set of all x such that x is in A or x is in B"
Properties of Union:
• Commutative: $A \cup B = B \cup A$
• Associative: $(A \cup B) \cup C = A \cup (B \cup C)$
• Identity: $A \cup \emptyset = A$
• Idempotent: $A \cup A = A$
• Domination: $A \cup U = U$ (where U is universal set)
• If $A \subseteq B$, then $A \cup B = B$
• Commutative: $A \cup B = B \cup A$
• Associative: $(A \cup B) \cup C = A \cup (B \cup C)$
• Identity: $A \cup \emptyset = A$
• Idempotent: $A \cup A = A$
• Domination: $A \cup U = U$ (where U is universal set)
• If $A \subseteq B$, then $A \cup B = B$
Example 1: Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Find $A \cup B$.
Solution:
List all elements that appear in either set (no repetition)
$A \cup B = \{1, 2, 3, 4, 5, 6\}$
Solution:
List all elements that appear in either set (no repetition)
$A \cup B = \{1, 2, 3, 4, 5, 6\}$
Example 2: Let $P = \{a, b, c\}$ and $Q = \{c, d, e, f\}$. Find $P \cup Q$.
Solution:
$P \cup Q = \{a, b, c, d, e, f\}$
(Note: 'c' appears in both sets but is listed only once)
Solution:
$P \cup Q = \{a, b, c, d, e, f\}$
(Note: 'c' appears in both sets but is listed only once)
Intersection of Sets
Intersection: The intersection of two sets $A$ and $B$ (denoted as $A \cap B$) is the set of all elements that are in BOTH $A$ AND $B$
Symbol: $\cap$ (cap symbol)
Keyword: "AND"
Disjoint Sets: Sets with no common elements ($A \cap B = \emptyset$)
Symbol: $\cap$ (cap symbol)
Keyword: "AND"
Disjoint Sets: Sets with no common elements ($A \cap B = \emptyset$)
Intersection Definition:
$$A \cap B = \{x | x \in A \text{ AND } x \in B\}$$
Read as: "A intersection B equals the set of all x such that x is in A and x is in B"
$$A \cap B = \{x | x \in A \text{ AND } x \in B\}$$
Read as: "A intersection B equals the set of all x such that x is in A and x is in B"
Properties of Intersection:
• Commutative: $A \cap B = B \cap A$
• Associative: $(A \cap B) \cap C = A \cap (B \cap C)$
• Identity: $A \cap U = A$ (where U is universal set)
• Idempotent: $A \cap A = A$
• Domination: $A \cap \emptyset = \emptyset$
• If $A \subseteq B$, then $A \cap B = A$
• Commutative: $A \cap B = B \cap A$
• Associative: $(A \cap B) \cap C = A \cap (B \cap C)$
• Identity: $A \cap U = A$ (where U is universal set)
• Idempotent: $A \cap A = A$
• Domination: $A \cap \emptyset = \emptyset$
• If $A \subseteq B$, then $A \cap B = A$
Example 1: Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Find $A \cap B$.
Solution:
List only the elements that appear in BOTH sets
$A \cap B = \{3, 4\}$
Solution:
List only the elements that appear in BOTH sets
$A \cap B = \{3, 4\}$
Example 2: Let $P = \{a, b, c\}$ and $Q = \{d, e, f\}$. Find $P \cap Q$.
Solution:
There are no common elements
$P \cap Q = \emptyset$ (empty set)
P and Q are disjoint sets
Solution:
There are no common elements
$P \cap Q = \emptyset$ (empty set)
P and Q are disjoint sets
Combined Operations
Example 1: Let $A = \{1, 2, 3, 4, 5\}$, $B = \{2, 4, 6, 8\}$, and $C = \{3, 4, 5, 6\}$
Find:
a) $A \cup B = \{1, 2, 3, 4, 5, 6, 8\}$
b) $A \cap B = \{2, 4\}$
c) $B \cap C = \{4, 6\}$
d) $A \cup C = \{1, 2, 3, 4, 5, 6\}$
e) $(A \cap B) \cup C = \{2, 4\} \cup \{3, 4, 5, 6\} = \{2, 3, 4, 5, 6\}$
f) $A \cap (B \cup C) = \{1, 2, 3, 4, 5\} \cap \{2, 3, 4, 5, 6, 8\} = \{2, 3, 4, 5\}$
Find:
a) $A \cup B = \{1, 2, 3, 4, 5, 6, 8\}$
b) $A \cap B = \{2, 4\}$
c) $B \cap C = \{4, 6\}$
d) $A \cup C = \{1, 2, 3, 4, 5, 6\}$
e) $(A \cap B) \cup C = \{2, 4\} \cup \{3, 4, 5, 6\} = \{2, 3, 4, 5, 6\}$
f) $A \cap (B \cup C) = \{1, 2, 3, 4, 5\} \cap \{2, 3, 4, 5, 6, 8\} = \{2, 3, 4, 5\}$
Example 2: Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, $A = \{1, 3, 5, 7, 9\}$, $B = \{2, 4, 6, 8, 10\}$
Find:
a) $A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} = U$
b) $A \cap B = \emptyset$ (disjoint sets)
c) $A' = \{2, 4, 6, 8, 10\} = B$
d) $B' = \{1, 3, 5, 7, 9\} = A$
e) $(A \cup B)' = U' = \emptyset$
f) $(A \cap B)' = \emptyset' = U$
Find:
a) $A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} = U$
b) $A \cap B = \emptyset$ (disjoint sets)
c) $A' = \{2, 4, 6, 8, 10\} = B$
d) $B' = \{1, 3, 5, 7, 9\} = A$
e) $(A \cup B)' = U' = \emptyset$
f) $(A \cap B)' = \emptyset' = U$
Venn Diagrams
Venn Diagram: A visual representation of sets using circles (or other shapes) inside a rectangle representing the universal set
Reading Venn Diagrams:
• Union ($A \cup B$): Entire area covered by both circles
• Intersection ($A \cap B$): Overlapping region of both circles
• Complement ($A'$): Everything in the rectangle except circle A
• Difference ($A - B$): Parts of A that don't overlap with B
• Disjoint sets: Circles that don't overlap
• Union ($A \cup B$): Entire area covered by both circles
• Intersection ($A \cap B$): Overlapping region of both circles
• Complement ($A'$): Everything in the rectangle except circle A
• Difference ($A - B$): Parts of A that don't overlap with B
• Disjoint sets: Circles that don't overlap
Venn Diagram Regions for Two Sets:
For sets A and B, there are typically 4 regions:
1. Only in A: $A - B$ or $A \cap B'$
2. Only in B: $B - A$ or $B \cap A'$
3. In both A and B: $A \cap B$
4. In neither A nor B: $(A \cup B)'$
For sets A and B, there are typically 4 regions:
1. Only in A: $A - B$ or $A \cap B'$
2. Only in B: $B - A$ or $B \cap A'$
3. In both A and B: $A \cap B$
4. In neither A nor B: $(A \cup B)'$
Interval Union and Intersection
Example 1: Find $[1, 5] \cup [3, 8]$ (Union of intervals)
Solution:
The union includes all numbers from 1 to 8
Answer: $[1, 8]$
Solution:
The union includes all numbers from 1 to 8
Answer: $[1, 8]$
Example 2: Find $[1, 5] \cap [3, 8]$ (Intersection of intervals)
Solution:
The intersection includes only the overlap (from 3 to 5)
Answer: $[3, 5]$
Solution:
The intersection includes only the overlap (from 3 to 5)
Answer: $[3, 5]$
Example 3: Find $(-\infty, 4) \cup [2, \infty)$
Solution:
This covers all real numbers
Answer: $(-\infty, \infty)$ or $\mathbb{R}$
Solution:
This covers all real numbers
Answer: $(-\infty, \infty)$ or $\mathbb{R}$
Example 4: Find $(1, 6) \cap [4, 9]$
Solution:
The overlap is from 4 (included) to 6 (not included)
Answer: $[4, 6)$
Solution:
The overlap is from 4 (included) to 6 (not included)
Answer: $[4, 6)$
Example 5: Find $(0, 3) \cap (5, 8)$
Solution:
These intervals don't overlap
Answer: $\emptyset$ (empty set)
Solution:
These intervals don't overlap
Answer: $\emptyset$ (empty set)
De Morgan's Laws
De Morgan's Laws for Sets:
Law 1: $(A \cup B)' = A' \cap B'$
The complement of a union is the intersection of complements
Law 2: $(A \cap B)' = A' \cup B'$
The complement of an intersection is the union of complements
Law 1: $(A \cup B)' = A' \cap B'$
The complement of a union is the intersection of complements
Law 2: $(A \cap B)' = A' \cup B'$
The complement of an intersection is the union of complements
Example: Let $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5, 6\}$
Verify De Morgan's First Law: $(A \cup B)' = A' \cap B'$
Left side:
$A \cup B = \{1, 2, 3, 4, 5, 6\}$
$(A \cup B)' = \{7, 8\}$
Right side:
$A' = \{5, 6, 7, 8\}$
$B' = \{1, 2, 7, 8\}$
$A' \cap B' = \{7, 8\}$ ✓
Both sides equal $\{7, 8\}$, so the law is verified!
Verify De Morgan's First Law: $(A \cup B)' = A' \cap B'$
Left side:
$A \cup B = \{1, 2, 3, 4, 5, 6\}$
$(A \cup B)' = \{7, 8\}$
Right side:
$A' = \{5, 6, 7, 8\}$
$B' = \{1, 2, 7, 8\}$
$A' \cap B' = \{7, 8\}$ ✓
Both sides equal $\{7, 8\}$, so the law is verified!
Quick Reference Guide
Set Notation Summary:
• $\in$ : element of
• $\notin$ : not an element of
• $\subseteq$ : subset of
• $\subset$ : proper subset of
• $\emptyset$ or $\{\}$ : empty set
• $U$ : universal set
• $A'$ or $A^c$ : complement of A
• $|A|$ or $n(A)$ : cardinality (number of elements)
• $\in$ : element of
• $\notin$ : not an element of
• $\subseteq$ : subset of
• $\subset$ : proper subset of
• $\emptyset$ or $\{\}$ : empty set
• $U$ : universal set
• $A'$ or $A^c$ : complement of A
• $|A|$ or $n(A)$ : cardinality (number of elements)
Set Operations:
• Union: $A \cup B = \{x | x \in A \text{ OR } x \in B\}$
• Intersection: $A \cap B = \{x | x \in A \text{ AND } x \in B\}$
• Complement: $A' = \{x \in U | x \notin A\}$
• Difference: $A - B = \{x | x \in A \text{ AND } x \notin B\}$
• Subset: $A \subseteq B$ if every element of A is in B
• Union: $A \cup B = \{x | x \in A \text{ OR } x \in B\}$
• Intersection: $A \cap B = \{x | x \in A \text{ AND } x \in B\}$
• Complement: $A' = \{x \in U | x \notin A\}$
• Difference: $A - B = \{x | x \in A \text{ AND } x \notin B\}$
• Subset: $A \subseteq B$ if every element of A is in B
Number of Subsets:
For a set with $n$ elements: Number of subsets = $2^n$
Number of proper subsets = $2^n - 1$
For a set with $n$ elements: Number of subsets = $2^n$
Number of proper subsets = $2^n - 1$
Interval Notation Quick Guide:
• $[a, b]$ : $a \leq x \leq b$ (closed)
• $(a, b)$ : $a < x < b$ (open)
• $[a, b)$ : $a \leq x < b$ (half-open)
• $(a, b]$ : $a < x \leq b$ (half-open)
• $(a, \infty)$ : $x > a$
• $[a, \infty)$ : $x \geq a$
• $(-\infty, b)$ : $x < b$
• $(-\infty, b]$ : $x \leq b$
• $(-\infty, \infty)$ : all real numbers
• $[a, b]$ : $a \leq x \leq b$ (closed)
• $(a, b)$ : $a < x < b$ (open)
• $[a, b)$ : $a \leq x < b$ (half-open)
• $(a, b]$ : $a < x \leq b$ (half-open)
• $(a, \infty)$ : $x > a$
• $[a, \infty)$ : $x \geq a$
• $(-\infty, b)$ : $x < b$
• $(-\infty, b]$ : $x \leq b$
• $(-\infty, \infty)$ : all real numbers
Important Properties:
• $A \cup A' = U$
• $A \cap A' = \emptyset$
• $(A')' = A$
• $U' = \emptyset$
• $\emptyset' = U$
• $A \cup \emptyset = A$
• $A \cap U = A$
• $A \cup U = U$
• $A \cap \emptyset = \emptyset$
• $A \cup A' = U$
• $A \cap A' = \emptyset$
• $(A')' = A$
• $U' = \emptyset$
• $\emptyset' = U$
• $A \cup \emptyset = A$
• $A \cap U = A$
• $A \cup U = U$
• $A \cap \emptyset = \emptyset$
De Morgan's Laws:
• $(A \cup B)' = A' \cap B'$
• $(A \cap B)' = A' \cup B'$
• $(A \cup B)' = A' \cap B'$
• $(A \cap B)' = A' \cup B'$
| Notation Type | Example | Meaning |
|---|---|---|
| Roster Form | $\{1, 2, 3, 4, 5\}$ | List all elements |
| Set-Builder | $\{x | x \in \mathbb{N}, x \leq 5\}$ | Describe property of elements |
| Interval | $[1, 5]$ | All numbers from 1 to 5 (inclusive) |
| Inequality | $1 \leq x \leq 5$ | x is between 1 and 5 |
Success Tips for Sets:
✓ Remember: Union = OR (combine all elements)
✓ Remember: Intersection = AND (only common elements)
✓ Complement includes everything NOT in the set
✓ Use [ ] for included endpoints, ( ) for excluded
✓ Always use ( ) with infinity symbols
✓ Practice converting between roster, set-builder, and interval notation
✓ Draw Venn diagrams to visualize relationships
✓ Check your answers by testing specific elements
✓ Remember: Union = OR (combine all elements)
✓ Remember: Intersection = AND (only common elements)
✓ Complement includes everything NOT in the set
✓ Use [ ] for included endpoints, ( ) for excluded
✓ Always use ( ) with infinity symbols
✓ Practice converting between roster, set-builder, and interval notation
✓ Draw Venn diagrams to visualize relationships
✓ Check your answers by testing specific elements