Study Notes on Sets and Set Notation

Introduction to Sets

  • Definition of Elements:

    • Elements or members are objects in a set.
    • Example: Each student in the class is an element or member of that set.

  • Characteristics of Sets:

    • A set must be well-defined.
    • The contents of a set must be clearly determined.
    • The order of elements in a set is not important.
    • Example: Seating arrangement can change without affecting the set of students.

Notation for Sets

  • Capital and Lowercase Letters:

    • Capital letters are used to name sets.
    • Lowercase letters are used for elements in the set.

  • Word Description:

    • Can designate a set using words.
    • Example: Capital W represents the set of the days of the week.

  • Roster Method:

    • This method lists the elements of a set.
    • Set notation uses curly brackets to enclose elements.
    • For example: {Washington, Adams, Jefferson, Madison, Monroe} represents the first five presidents of the United States.

  • Comma Usage:

    • Commas are used to separate elements in a set.

Examples of Set Descriptions

  • Word Description Example:

    • Description: The set of the first five presidents of the United States.

  • Current US Coins:

    • Set of US coins with a value of less than a dollar.
    • Includes penny, nickel, dime, quarter, half dollar.
    • Important note: Cannot include a dollar because it is not less than a dollar.

Types of Set Notations

  • Set Builder Notation:

    • Combines elements and conditions to describe a set.
    • Example Formula:
    • W = { x | x is a day of the week }
    • Expression of conditions is indicated by the vertical line (|), translated to “such that.”

  • Detailed Example in Set Builder Notation:

    • Q = { x | x ≥ 10 }
    • Converts to a word description: Set of all whole numbers greater than or equal to 10.
    • Roster example would begin with {11, 12, 13…} representing whole numbers greater than 10.

Empty Set

  • Definition:
    • The empty set, also referred to as the null set, is represented by:
    • Curly brackets with nothing inside: {}
    • A circle with a line through it: Ø
    • Importance: The empty set contains no elements, not even a space.

Elements and Sets

  • Element Definition:

    • A variable representing an object in a set. Example: Let x represent an unknown element.

  • Set Notation for Specific Conditions:

    • Example: A = { x | x is a month starting with 'M' }
    • Roster Method Result: {March, May}

Properties of Sets

  • Cardinal Number:

    • Refers to the number of distinct elements in a set.
    • Denoted as n(A) or similar notation.
    • Example: n({1,2,3,4}) = 4.
    • Repeating elements do not add to cardinality.
    • Example: n({1,1,2,3}) = 3.

  • Equivalent Sets:

    • Two sets are equivalent if they have the same number of elements, but not necessarily the same elements.
    • Example: Set A and Set B both contain 37 students, hence they are equivalent.

  • Finite and Infinite Sets:

    • Finite set: Cardinality is a natural number or zero.
    • Infinite set: Cardinality is not a natural number.

Equal Sets

  • Definition of Equal Sets:
    • Two sets are equal if they contain exactly the same elements, regardless of order or repetition.
    • Order does not matter: {1,2,3} = {3,2,1}.
    • Example of Equality:
    • Set A = {2,4,6}, Set B = {4,2,6} are equal.

Conclusion

  • Revisit key concepts such as the definitions of sets, the importance of notation, and the different types of set descriptions.
  • Review the significance of cardinality and conditions that determine element inclusion in sets.
  • Understand the distinctions between equivalent sets and equal sets, ensuring clarity on terminology and their mathematical implications.