Sets & Numerical Sysrens

Sets and Numeral Systems

2.1 Sets

Overview

• Sets are fundamental collections of objects, which include numbers, letters, or even other sets.

• Elements/Members: Items contained within a set are called elements or members.

• Notation: Sets are typically denoted by capital letters (e.g., A, B).

Definitions of a Set

• Definition: A set is defined as a collection of objects.

Common Methods to Describe a Set

1. Listing Method: A set can be described by explicitly listing its elements.

   • Example: A = ext{{}}{1, 2, 3, 4, 5 ext{{}}}

2. Set-Builder Notation: This defines sets by a property that its members must satisfy.

   • Example: B = ext{{}}{x | x ext{{ is a whole number from }} 1 ext{{ to }} 5 ext{{}}}

Key Types of Sets

• Null Set: Denoted by ext{{}} or ext{{}} , denoting an empty set with no elements.

• Equality of Sets: Two sets are considered equal, denoted as $A = B$, if they contain the same elements, irrespective of order.

Cardinality of a Set

• Definition of Cardinality: The cardinality of a set refers to the number of unique elements it contains.

• Finite Set: A set with limited elements.

• Infinite Set: A set with unlimited elements.

• Example: The null set has a cardinality of 0.

Examples of Finding Cardinality

1. Given a set A = ext{{}}{a, b, c, d ext{{}}} :

   • Finite set with cardinality = 4

2. Set B = ext{{}}{1, 2, 3, 4, 5, 6 ext{{}}} :

   • Finite set with cardinality = 6

3. The null set C = ext{{}} :

   • Finite set with cardinality = 0

4. Set D = ext{{}}{x | x ext{{ is an odd number ext{{}}}} :

   • Infinite set due to infinite odd numbers.

2.2 Whole Numbers and Natural Numbers

Natural Numbers

• Definition of Natural Numbers: Set of all cardinalities of nonempty finite sets, also called counting numbers.

• Set Representation: Natural numbers are represented as ext{{}}{1, 2, 3, 4, 5, 6, ext{{…}} ext{{}}}

Whole Numbers

• Definition of Whole Numbers: Set of all cardinalities of finite sets, including zero.

• Set Representation: Whole numbers are represented as ext{{}}{0, 1, 2, 3, 4, 5, 6, ext{{…}} ext{{}}}

Comparison of Natural Numbers and Whole Numbers

• Key Difference: Natural numbers do not include zero, while whole numbers do include zero.

  • ext{{Natural Numbers}} = ext{{}}{1, 2, 3, … ext{{}}}

  • ext{{Whole Numbers}} = ext{{}}{0, 1, 2, 3, … ext{{}}}

2.3 Comparison, Equality, and Equivalence of Sets

Equivalent Sets

• Definition: Two sets A and B are equivalent, denoted $A ilde B$, if they have the same number of elements, even if the elements are different.

  • Example: Sets with cardinality 4, such as A = ext{{}}{a, b, c, d ext{{}}} and B = ext{{}}{e, f, g, h ext{{}}} are equivalent.

Equality of Sets

• Definition: Sets A and B are equal $A = B$ if they contain precisely the same elements.

  • Example: Sets C = ext{{}}{a, e, i, o, u ext{{}}} and D = ext{{}}{a, e, i, o, u ext{{}}} are equal.

Subsets and Proper Subsets

• Definition of Subset: Set A is a subset of set B, denoted $A ext{ C } B$, if every element of A is also an element of B.

• Definition of Proper Subset: Set A is a proper subset of B, denoted $A ext{ C } B$, if A is a subset of B but not equal to B.

Examples of Subsets

1. Set A = ext{{}}{1, 2, 3 ext{{}}} has subsets ext{{}}{ ext{{}}, ext{{}}{1 ext{{}}}, ext{{}}{2 ext{{}}}, ext{{}}{3 ext{{}}}, ext{{}}{1, 2 ext{{}}}, ext{{}}{1, 3 ext{{}}}, ext{{}}{2, 3 ext{{}}}, ext{{}}{1, 2, 3 ext{{}}} (total of 8 subsets).

2. A set with n elements has a total of $2^n$ subsets.

Inequalities Between Sets

• Definition of Inequality for Whole Numbers: If $m$ is the cardinality of set A and $n$ is the cardinality of set B:

  • m ext{ <= } n is expressed as $A C B$.

  • m < n is expressed as $A C B$, indicating a proper subset.

• Example: Let A contain 3 elements and B contain 7, then the relationship 3 < 7 holds.

Classroom Activities

• Encourage students to consider practical examples to visualize set cardinalities, such as matching cookies with bags to demonstrate equivalent sets.

• Facilitate activities where students derive subsets from given sets to solidify the concept.

• Use set-builder notation and listing method interchangeably to enhance comprehension of set definitions.