CTJan27 Online JMSS Exam Prep - Set Theory

Introduction to Sets

A set is a collection of distinct objects, called elements or members. We use curly braces {} to denote a set.

Set Membership

• The symbol $∈$ means "is an element of". For example, if $A = \{1, 2, 3\}$, then $2 ∈ A$.
• The symbol $∉$ means "is not an element of". For example, using the same set $A$, $4 ∉ A$.

Subsets

• Set $B$ is a subset of set $A$ if every element of $B$ is also an element of $A$. We write this as $B ⊆ A$.
• For example, if $A = \{1, 2, 3, 4\}$ and $B = \{1, 3\}$, then $B ⊆ A$.

Set Operations

Let $A = \{1, 3, 5, 7\}$ and $B = \{5, 7, 9, 11\}$.

• Intersection ($∩$): The set of elements common to both sets.
  • $A ∩ B = \{5, 7\}$

• Union ($∪$): The set of all elements that are in either set (or both). Duplicates are not listed.
  • $A ∪ B = \{1, 3, 5, 7, 9, 11\}$

• Set Difference ($ \setminus $): The set of elements in the first set but not in the second set.
  • $A \setminus B = \{1, 3\}$ (Elements in A but not in B)
  • $B \setminus A = \{9, 11\}$ (Elements in B but not in A)

Special Sets

• Empty Set ($∅$): A set with no elements, also written as \{\}.
• Disjoint Sets: Two sets are disjoint if they have no elements in common, meaning their intersection is the empty set. For example, if $C = \{1, 2\}$ and $D = \{3, 4\}$, then $C ∩ D = ∅$.

Sets of Numbers

• Natural Numbers ($N$): The counting numbers. $N = \{1, 2, 3, ...\}$
• Integers ($Z$): Whole numbers and their negatives, including zero. $Z = \{..., -2, -1, 0, 1, 2, ...\}$
• Rational Numbers ($Q$): Numbers that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q ≠ 0$. This includes terminating and repeating decimals (e.g., $0.5, -3, 0.333...$).
• Real Numbers ($R$): All rational and irrational numbers. Irrational numbers cannot be written as a simple fraction (e.g., $π, \sqrt{2}$).

Questions for: Determining set membership using ∈ and ∉

Questions for: Identifying subsets using the ⊆ symbol

Questions for: Calculating the intersection of two sets (A ∩ B)

Questions for: Calculating the union of two sets (A ∪ B)

Questions for: Finding the set difference (A \ B)

Questions for: Identifying disjoint sets and using the empty set symbol (∅)

Questions for: Classifying numbers into sets (N, Z, Q, R)

Questions for: Describing sets using set-builder notation

A set is a collection of distinct objects, called elements or members. We can describe a set in two primary ways: the Roster Method and Set-Builder Notation.

Roster Method

This method involves listing all the elements of the set, separated by commas, within curly braces {}.

For example, the set of the first five positive integers is A = {1, 2, 3, 4, 5}.

This method is convenient for sets with a small number of elements.

Set-Builder Notation

When it's not practical to list all elements (especially for infinite sets), we use set-builder notation. This notation describes the properties that an element must have to be included in the set. The general form is:

{ x | P(x) } or { x : P(x) }

This is read as: "The set of all elements x such that the property P(x) is true."

Let's break down its components:

• { } : The curly braces indicate we are defining a set.
• x : A variable representing any element in the set.
• | or : : The vertical bar or colon is read as "such that".
• P(x) : The property or rule that the element x must satisfy. This often involves specifying which larger set the elements come from (e.g., natural numbers $\mathbb{N}$, integers $\mathbb{Z}$, real numbers $\mathbb{R}$).

Examples:

1. Roster: B = {5, 6, 7, 8} Set-Builder: B = { x | x ∈ ℕ, 4 < x < 9 } or B = { x | x is an integer and 5 ≤ x ≤ 8 }.

2. Description: The set of all even integers. Roster (partial): C = {..., -4, -2, 0, 2, 4, ...} Set-Builder: C = { x | x = 2n, n ∈ ℤ }. This means "the set of all elements x such that x is the product of 2 and an integer n."

3. Description: The set of all real numbers greater than 10. Set-Builder: D = { x | x ∈ ℝ, x > 10 }. This describes an infinite set that cannot be listed using the roster method.

Questions for: Interpreting sets written in set-builder notation

Set-builder notation is a mathematical shorthand for describing a set by stating the properties that its members must satisfy. It's particularly useful for sets with many elements or infinite sets.

The Structure

The general form of set-builder notation is:

{ variable | condition(s) } or { variable : condition(s) }

Let's break down the components:

• { } (Braces): These indicate that we are defining a set.
• variable: This is a placeholder (like $x$, $y$, or $n$) that represents any element in the set.
• | or :: This is a separator that is read as "such that". It separates the variable from the rule.
• condition(s): This is the rule or property that an element must satisfy to be included in the set. It often involves specifying which larger set the elements come from (like natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$) and any additional constraints.

Examples

Example 1: A set of Integers

Consider the set $A = \{ x : x \in \mathbb{Z} \text{ and } -2 < x \le 3 \}$.

• Read as: "The set of all elements $x$ such that $x$ is an integer and $x$ is greater than -2 and less than or equal to 3."
• To list the elements (roster form): We need to find all integers that fit the condition. These are -1, 0, 1, 2, and 3.
• So, in roster form: $A = \{-1, 0, 1, 2, 3\}$.

Example 2: A set with a formula

Consider the set $B = \{ 2n + 1 : n \in \mathbb{N} \}$. (Remember, $\mathbb{N} = \{1, 2, 3, ...\}$).

• Read as: "The set of all elements of the form $2n+1$ such that $n$ is a natural number."
• To list the elements: We substitute values of $n$ from the natural numbers into the formula $2n+1$.
  • If $n=1$, then $2(1) + 1 = 3$.
  • If $n=2$, then $2(2) + 1 = 5$.
  • If $n=3$, then $2(3) + 1 = 7$.
• So, in roster form: $B = \{3, 5, 7, 9, ...\}$. This is the set of odd integers greater than 1.

Example 3: A set with a verbal description

Consider the set $C = \{ p | p \text{ is a prime number and } p < 15 \}$.

• Read as: "The set of all elements $p$ such that $p$ is a prime number and $p$ is less than 15."
• To list the elements: We list all the prime numbers that are smaller than 15.
• So, in roster form: $C = \{2, 3, 5, 7, 11, 13\}$.

To interpret a set in set-builder notation, you must first identify the type of numbers you are working with, and then carefully apply all the conditions to find the elements that belong in the set.

Questions for: Converting from interval notation to an inequality

Interval notation is a convenient way to represent a set of numbers between two endpoints. Converting this notation into an inequality is a fundamental skill. The key is to understand what the different types of brackets mean.

Open Intervals: Parentheses ( )

Parentheses, like (a, b), indicate an open interval. This means the endpoints a and b are not included in the set. In inequalities, this corresponds to the 'less than' ($<$) and 'greater than' ($>$) symbols.

• Example: The interval (3, 8) represents all the real numbers strictly between 3 and 8.
• As an inequality: $3 < x < 8$

Closed Intervals: Square Brackets [ ]

Square brackets, like [a, b], indicate a closed interval. This means the endpoints a and b are included in the set. In inequalities, this corresponds to the 'less than or equal to' ($\le$) and 'greater than or equal to' ($\ge$) symbols.

• Example: The interval [-2, 5] represents all the real numbers between -2 and 5, including -2 and 5.
• As an inequality: $-2 \le x \le 5$

Half-Open or Half-Closed Intervals

Sometimes, an interval includes one endpoint but not the other. These are called half-open or half-closed intervals and use a combination of a parenthesis and a square bracket.

• Example 1: The interval (1, 6] includes 6 but not 1.

• As an inequality: $1 < x \le 6$

• Example 2: The interval [-4, 0) includes -4 but not 0.

• As an inequality: $-4 \le x < 0$

Intervals with Infinity (∞)

The symbols ∞ (infinity) and -∞ (negative infinity) are used to show that an interval continues forever in a positive or negative direction. Since infinity is a concept and not a specific number, it is always paired with a parenthesis ( or ).

• Example 1: The interval (5, ∞) represents all real numbers greater than 5.

• As an inequality: $x > 5$

• Example 2: The interval (-∞, 10] represents all real numbers less than or equal to 10.

• As an inequality: $x \le 10$

Questions for: Converting from an inequality to interval notation

Examples

Let's look at a few examples to see this in action:

1. Inequality: $-2 \le x \le 3$
   • Explanation: The set includes all numbers between -2 and 3, including -2 and 3. Both endpoints are included, so we use square brackets.
   • Interval Notation: $[-2, 3]$
2. Inequality: $-3 < x \le 4$
   • Explanation: The set includes all numbers between -3 and 4. It does not include -3 (so we use a parenthesis) but does include 4 (so we use a square bracket).
   • Interval Notation: $(-3, 4]$
3. Inequality: $x > -3$
   • Explanation: The set includes all numbers greater than -3, extending to positive infinity. It does not include -3.
   • Interval Notation: $(-3, \infty)$
4. Inequality: $x \le 5$
   • Explanation: The set includes all numbers less than or equal to 5, extending from negative infinity. It does include 5.
   • Interval Notation: $(-\infty, 5]$

Questions for: Representing intervals on a number line

In mathematics, an interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers $x$ satisfying $0 \le x \le 1$ is an interval. There are different ways to write intervals, and a common way is to show them on a number line.

Key Concepts

1. Endpoints: These are the numbers at the beginning and end of an interval.
2. Included vs. Excluded Endpoints: Sometimes an endpoint is part of the interval, and sometimes it is not.
   • A closed circle (•) on a number line means the endpoint is included in the interval. This corresponds to the symbols $\le$ (less than or equal to) or $\ge$ (greater than or equal to).
   • An open circle (◦) on a number line means the endpoint is not included. This corresponds to the symbols %%%MATH_BLOCK_4%%%lt;$ (less than) or %%%MATH_BLOCK_5%%%gt;$ (greater than).

Interval Notation

Interval notation is a concise way to describe an interval. It uses parentheses and square brackets.

• Parentheses ( ): Used when an endpoint is not included (matches an open circle ◦).
• Square Brackets [ ]: Used when an endpoint is included (matches a closed circle •).

Let's assume $a$ and $b$ are real numbers with $a < b$.

Infinite Intervals

Intervals can also extend forever in one or both directions. We use the infinity symbol, $\infty$, and the negative infinity symbol, $-\infty$, for this. Infinity is not a number, so we always use a parenthesis next to $\infty$ or $-\infty$.

Example:

Represent the interval [-2, 3) on a number line.

• -2 has a square bracket [, so it's included. We use a closed circle (•) at -2.
• 3 has a parenthesis ), so it's not included. We use an open circle (◦) at 3.
• We shade the line between -2 and 3.

This interval corresponds to the set of all real numbers $x$ such that $-2 \le x < 3$.

Questions for: Finding the union and intersection of intervals

Introduction to Intervals

An interval is a set containing all real numbers between two given numbers, called the endpoints. We use different brackets to indicate whether the endpoints are included.

• Parentheses ( ): The endpoint is not included. For example, $(2, 5)$ means all numbers strictly between 2 and 5.
• Square brackets [ ]: The endpoint is included. For example, $[2, 5]$ means all numbers between 2 and 5, including 2 and 5.

You can mix these, like $(2, 5]$ which includes 5 but not 2. The symbols $\infty$ (infinity) and $-\infty$ (negative infinity) are used for intervals that extend forever in one direction and always use parentheses.

Intersection of Intervals ($ \cap $)

The intersection of two intervals, A and B, written as $A \cap B$, is the set of all numbers that are in both interval A and interval B. Think of it as the "overlap" between the two intervals.

How to find the intersection:

1. Draw both intervals on the same number line.
2. The intersection is the region where the two intervals overlap.

Example 1: Find $[-1, 3] \cap (0, 4)$.

• Interval A is $[-1, 3]$: all numbers from -1 to 3, including -1 and 3.
• Interval B is $(0, 4)$: all numbers from 0 to 4, not including 0 and 4.

On a number line, the overlap starts just after 0 and ends at 3.

• Is 0 in the overlap? No, because it's not in $(0, 4)$.
• Is 3 in the overlap? Yes, because it's in both $[-1, 3]$ and $(0, 4)$.

Therefore, $[-1, 3] \cap (0, 4) = (0, 3]$.

Example 2: Find $(-\infty, 2) \cap [5, \infty)$. These two intervals do not overlap at all. The first one contains numbers less than 2, and the second contains numbers greater than or equal to 5. So, their intersection is the empty set, denoted by $\emptyset$.

Union of Intervals ($ \cup $)

The union of two intervals, A and B, written as $A \cup B$, is the set of all numbers that are in interval A, or in interval B, or in both. Think of it as "combining" or "uniting" the two intervals.

How to find the union:

1. Draw both intervals on the same number line.
2. The union is the total region covered by either of the intervals.

Example 1: Find $[-1, 3] \cup (0, 4)$.

• Interval A is $[-1, 3]$.
• Interval B is $(0, 4)$.

If we combine these, the resulting region starts at -1 (included) and goes all the way to 4 (not included). Therefore, $[-1, 3] \cup (0, 4) = [-1, 4)$.

Example 2: Find $(-\infty, 0] \cup [2, 5]$. These two intervals are separate. When we combine them, they remain separate. We write the union by simply connecting them with the union symbol. Therefore, the union is $(-\infty, 0] \cup [2, 5]$. It cannot be simplified into a single interval.