Number Sets and Set Notation Principles

Set Notation and Basic Terminology

• Definition of a Set: A set is a collection of numbers or objects.
• Notation Examples:
  • If $V$ is the set of all vowels, then $V = \{vowels\} = \{a, e, i, o, u\}$.
  • If $E$ is the set of all even numbers, then $E = \{even \text{ } numbers\} = \{2, 4, 6, 8, 10, 12, \dots\}$.
• Membership Symbols:
  • The symbol $\in$ means "is an element of."
  • The symbol $\notin$ means "is not an element of."
  • Example: For the set $E = \{2, 4, 6, 8, 10, 12, \dots\}$, we can state $6 \in E$ but $11 \notin E$.
• The Empty Set: The set $\emptyset$ (or $\text{\{ \}}$) is called the empty set and contains no elements.
• General Examples:
  • $A = \{1, 2, 3, 4\}$. We can say $3 \in A$ and $5 \notin A$.
  • $B = \{\text{Rashid, Laith, Hussein}\}$. Here, $Rashid \in B$ and $Hussein \in B$, but $Omar \notin B$.

Counting Elements and Set Classification

• Cardinality: The number of elements in a set $S$ is written as $n(S)$.
  • Example: For set $B = \{\text{Rashid, Laith, Hussein}\}$, $n(B) = 3$.
• Finite Sets: A set which contains a finite (countable) number of elements is called a finite set.
  • Example: The set of vowels $V$ has 5 elements. $V$ is a finite set, and $n(V) = 5$.
• Infinite Sets: A set which contains an infinite number of elements is called an infinite set.
  • Example: The set of even numbers $E$ is an infinite set.

Special Number Sets

• Natural Numbers (Counting Numbers): $\mathbb{N} = \{0, 1, 2, 3, 4, 5, 6, 7, \dots\}$.
• Integers: $\mathbb{Z} = \{0, \pm 1, \pm 2, \pm 3, \pm 4, \dots\}$.
• Positive Integers: $\mathbb{Z}^+ = \{1, 2, 3, 4, 5, 6, 7, \dots\}$.
• Rational Numbers ($\mathbb{Q}$): The set of all numbers which can be written in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.
  • Examples: $15$, $10 = \frac{10}{1}$, $0.5 = \frac{1}{2}$, and $-\frac{3}{4}$.
• Irrational Numbers: These are numbers that cannot be represented in rational form ($\frac{p}{q}$).
  • Radicals/Surds: Examples include $\sqrt{2}$ and $\sqrt{7}$.
  • Mathematical Constants: $\pi \approx 3.14159265$ is irrational.
  • Non-recurring/Non-terminating Decimals: Decimal numbers that neither terminate nor repeat are irrational.
• Real Numbers ($\mathbb{R}$): The set of all numbers that can be placed on the number line. $\mathbb{R}$ includes all rational and irrational numbers.
• Non-Real Numbers: These are numbers that cannot be written in decimal form or placed on a number line.
  • Examples: $\sqrt{-25}$, $\frac{5}{0}$, and $\sqrt{-2}$.

Exercises on Writing Set Notation

• Write the following using symbols:
  • 8 is an element of set $P$: $8 \in P$
  • $k$ is not an element of set $S$: $k \notin S$
  • 14 is not an element of the set of all odd numbers: $14 \notin \text{odd numbers}$
  • There are 9 elements in set $Y$: $n(Y) = 9$

Verification of Set Membership (True or False)

• $3 \in \mathbb{Z}^+$: True
• $6 \in \mathbb{Z}$: True
• $\frac{2}{3} \in \mathbb{Q}$: True
• $\sqrt{2} \notin \mathbb{Q}$: True (since it is irrational)
• $-\frac{1}{4} \notin \mathbb{Q}$: False (it is rational)
• $2 \frac{1}{3} \in \mathbb{Z}$: False (it is a fraction, not an integer)
• $0.3684 \in \mathbb{R}$: True
• $\frac{1}{0.1} \in \mathbb{Z}$: True (Evaluation: $\frac{1}{0.1} = 10$, and $10 \in \mathbb{Z}$)

Listing Elements and Set Evaluation

Rules for Listing Elements:

• Order of elements is not important.
• Repetition of elements is NOT allowed.
• Always use curly brackets $\text{\{ \}}$.

Set Analysis Examples:

1. Set $A = \{\text{factors of 6}\}$:
   • Elements: $A = \{1, 2, 3, 6\}$
   • Type: Finite Set
   • Cardinality: $n(A) = 4$
2. Set $B = \{\text{multiples of 6}\}$:
   • Elements: $B = \{6, 12, 18, 24, \dots\}$
   • Type: Infinite Set
3. Set $C = \{\text{factors of 17}\}$:
   • Elements: $C = \{1, 17\}$
   • Type: Finite Set
   • Cardinality: $n(C) = 2$
4. Set $D = \{\text{multiples of 17}\}$:
   • Elements: $D = \{17, 34, 51, \dots\}$
   • Type: Infinite Set
5. Set $E = \{\text{prime numbers less than 20}\}$:
   • Elements: $E = \{2, 3, 5, 7, 11, 13, 17, 19\}$
   • Type: Finite Set
   • Cardinality: $n(E) = 8$

Understanding Subsets

• Definition: Suppose $A$ and $B$ are two sets. $A$ is a subset of $B$ (written as $A \subseteq B$) if every element of $A$ is also an element of $B$.
• Fundamental Principles:
  • If $E = \{\text{even numbers}\}$, then $E \subseteq \mathbb{Z}$.
  • The empty set $\emptyset$ is a subset of every set.

Subset Examples:

• If $A = \{2, 4, 6, 8, 10, 12\}$ and $B = \{2, 6, 12\}$, then $B \subseteq A$.
• If $A = \{2, 4, 6, 8, 10, 12\}$ and $C = \{2, 6, 13\}$, then $C \nsubseteq A$ because $13 \notin A$.
• Suppose $A = \{1, 2, 3, 4, 5, 6, 7\}$, $B = \{2, 3, 5\}$, and $C = \{3, 5, 8\}$.
  • $B \subseteq A$ is True.
  • $C \subseteq A$ is False (due to the element 8).

Subset Verification Practice

• Example a: $A = \{2, 5, 6\}$, $B = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Result: $A \subseteq B$.
• Example b: $A = \{4, 8, 11, 12\}$, $B = \{2, 4, 6, 8, 10, 12, 14, 16\}$. Result: $A \nsubseteq B$ (11 is in $A$ but not in $B$).
• Example c: $A = \emptyset$, $B = \{1, 4, 7, 10\}$. Result: $A \subseteq B$ (Empty set is a subset of every set).
• Example d: $A = \{5, 10, 15, 20, 25, 30\}$, $B = \{10, 15, 20\}$. Result: $A \nsubseteq B$ (but clearly $B \subseteq A$).
• Example e: $A = \{6, 7, 8\}$, $B = \mathbb{N}$. Result: $A \subseteq B$.

Homework Tasks

Given the following sets:

• $P = \{\text{prime numbers less than 10}\}$ (Analysis: $P = \{2, 3, 5, 7\}$)
• $Q = \{\text{multiples of 3 less than 20}\}$ (Analysis: $Q = \{3, 6, 9, 12, 15, 18\}$)
• $R = \{3, 5, 7\}$
• $S = \{\text{multiples of 6 less than 20}\}$ (Analysis: $S = \{6, 12, 18\}$)

Determine if the following are True or False:

• $P \subseteq Q$
• $R \subseteq P$
• $R \subseteq S$
• $S \subseteq Q$