Introduction to Number Systems

Introduction to Number Systems

Definition of a Set

A set is defined as any collection of objects specified in such a way that we can determine whether a given object is or is not a member of the collection.

Key Points

• Sets are denoted by capital letters such as 𝐴, 𝐵, 𝐶, etc.

• Each object within a set is referred to as an element or member of that set.

• Sets can be described in the following ways:

  • Roster Notation: Listing all elements between braces.

  • Example: 𝐴 = {1, 2, 3}.

  • The notation 2 ∈ 𝐴 indicates that 2 is an element of set 𝐴, while 4 ∉ 𝐴 indicates that 4 is not a member of this set.

  • Set-builder Notation: Enclosing a rule within braces.

  • Example: 𝐵 = {𝑥 ∈ ℕ | 1 ≤ 𝑥 ≤ 5}.

  • This notation expresses that 𝐵 includes all natural numbers 𝑥 that satisfy the condition of being between 1 and 5, which can also be written in Roster Notation as 𝐵 = {1, 2, 3, 4, 5}.

• A set can be classified as finite or infinite:

  • A finite set has a countable number of elements; for instance, 𝐴 = {1, 2, 3} is finite.

  • An infinite set does not end; for example, 𝐶 = {2, 4, 6, 8, 10, …} is infinite.

• When listing a set, elements are not repeated.

Special Types of Sets

• Empty Set (Null Set): A set with no elements, denoted by either {} or ∅.

Equality of Sets

• Two sets 𝐴 and 𝐵 are equal if they contain the same elements.

  • For example, if 𝐴 = {1, 2, 3} and 𝐵 = {3, 1, 2}, then 𝐴 = 𝐵 since both sets comprise the same elements.

Types of Numbers

• Natural numbers (ℕ): The set of positive integers {1, 2, 3, …}.

• Integers (ℤ): The set of whole numbers …, -3, -2, -1, 0, 1, 2, 3, …

• Rational numbers (ℚ): Numbers that can be expressed in the form $rac{p}{q}$, where $p, q ext{ in } ℤ$ and $q ≠ 0$.

  • Examples include -$rac{1}{2}$, 0.1, $rac{1}{3}$, 2, -10, 0, etc.

• Irrational numbers (ℝ): Numbers that cannot be written as the fraction of integers, such as $ext{√2}, ext{π}, e$.

• Real numbers (ℝ): The union of rational and irrational numbers (ℚ ∪ ℝ).

Note

• The default set for this course will be the set of real numbers: $U = ℝ$.

Relational Symbols

• Equal to ( = ): $x = y$ if $x$ and $y$ are the same.

• Not equal to ( ≠ ): $x ≠ y$ if $x$ and $y$ are not the same.

• Less than ( < ): x < y when $x$ is smaller than $y$.

• Greater than ( > ): x > y when $x$ is larger than $y$.

• Less than or equal to ( ≤ ): $x ≤ y$ when $x$ is either smaller than or equal to $y$.

• Greater than or equal to ( ≥ ): $x ≥ y$ when $x$ is either larger than or equal to $y$.

• Implies ( ⇒ ): $P ⇒ Q$ means that if $P$ is true, then $Q$ is true.

• If and only if ( ⇔ ): $P ⇔ Q$ means that if $P$ is true, then $Q$ is also true, and vice versa.

Set Operations

1. Union ( ∪ ): The union of two sets includes elements in either set:

   • $x ∈ A ∪ B$ if $x ∈ A$ or $x ∈ B$.

   • Example:

     • Let 𝐴 = {1, 2, 3}, 𝐵 = {3, 4, 5}, 𝐶 = {6}.

     • Then, $A ∪ B = {1, 2, 3, 4, 5}$ and $B ∪ C = {3, 4, 5, 6}$.

2. Intersection ( ∩ ): The intersection of two sets contains elements common to both sets:

   • $x ∈ A ∩ B$ if $x ∈ A$ and $x ∈ B$.

   • Example:

     • Using previous sets, $A ∩ B = {3}$, while $B ∩ C = ∅$.

3. Difference ( − ): The difference of two sets includes elements that are in one set but not the other:

   • $x ∈ (A − B)$ is true if $(x ∈ A)$ and $(x ∉ B)$.

   • Example:

     • For sets 𝐴 and 𝐵 as stated previously, $A − B = {1, 2}$ and $B − C = {3, 4, 5}$.

4. Subset ( ⊆ ): A set 𝐴 is a subset of set 𝐵 if every element of 𝐴 is also an element of 𝐵:

   • Example:

     • Let 𝐴 = {1, 2, 3} and 𝐵 = {1, 2, 3, 4, 5}. Thus, $A ⊆ B$, but $B ⊈ A$.

Important Notes on Subsets

• Two sets 𝐴 and 𝐵 are equal if both $A ⊆ B$ and $B ⊆ A$ hold true.

• The relationship between number sets: $ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ$.

• The union of rational numbers with irrational numbers results in the real numbers: $ℚ ∪ I = ℝ$.

• The Universal Set (U) is the set of all elements under consideration in a particular problem. All other sets must be subsets of $U$.

1. Complement: The complement of a set 𝐴 (relative to the Universal Set 𝑈), denoted as 𝐴′ or 𝐴ᶜ, contains all elements in 𝑈 that are not in 𝐴:

   • Symbolically, this is represented as 𝐴′ = {𝑥 ∈ 𝑈 | 𝑥 ∉ 𝐴}.

   • Example:

     • If $U = {1, 2, 3, 4, 5}$ and $A = {1, 2, 3}$, then $A' = {4, 5}$.

   • Important property: $A ∪ A′ = U$.

2. Proper Subset ( ⊂ ): A set 𝐴 is a proper subset of set 𝐵 if every element of 𝐴 is contained in 𝐵, and there exists at least one element in 𝐵 that is not in 𝐴:

   • Example:

     • With 𝐴 = {1, 2, 3} and 𝐵 = {1, 2, 3, 4, 5}, then it holds that $A ⊂ B$.