Sets in Mathematics Quiz

Sets in Mathematics

• Sets are collections of distinct objects, called elements or members.
• Elements can be numbers, symbols, points, lines, shapes, variables, or other sets.
• Sets can be finite or infinite.
• The empty set is a unique set with no elements.
• A set with one element is a singleton.

Defining Sets

• Extensionality: Two sets are equal if and only if they have the same elements.
• Roster (Enumeration) Notation: A set is defined by listing its elements within curly brackets, separated by commas. Order doesn't matter.
  • Example: {4, 2, 1, 3} = {1, 2, 3, 4}
  • Ellipsis (...) can be used to abbreviate lists with many elements or an obvious pattern. Example: First thousand positive integers: {1, 2, 3, ..., 1000}
  • Infinite sets: an ellipsis in roster notation signifies the list continues forever. {..., -3, -2, -1, 0, 1, 2, 3,...} represents all integers.
• Semantic Definition: A set is defined by a rule that determines its elements. - Examples: The first four positive integers. The colors of the French flag.
• Set-builder Notation: Defines a set as a selection from a larger set based on a condition. Example: {n | n is an integer, 0 ≤ n ≤ 19} (read as "the set of all n such that n is an integer and is between 0 and 19, inclusive").
• Classifying Definitions:
  • Intensional: Uses a rule to determine membership.
  • Extensional: Describes a set by listing all its elements.
  • Ostensive: Describes a set by giving examples.

Set Membership

• Membership symbol: x ∈ B ("x belongs to B", "x is in B")
• Non-membership symbol: x ∉ B ("x is not in B")
• Example: 4 ∈ {1, 2, 3, 4} and 20 ∉ {1, 2, ..., 19}.

Special Sets

• Empty Set: The unique set with no elements. Notation: ∅, {}.
• Singleton Set: A set with exactly one element. Example: {x} (where x is the element)
  • A singleton set and its element are distinct concepts (like a box containing a hat and the hat itself)