1.1 Basic Set Concepts

Set theory forms the foundation of mathematical reasoning. It's all about organizing and categorizing objects into collections called sets. Understanding sets helps us make sense of relationships between different groups of things.

Sets can be finite or infinite, equal or equivalent. We use special symbols and notations to describe sets and their relationships. These concepts are crucial for solving problems in math and beyond.

Fundamentals of Set Theory

Components of sets

Top images from around the web for Components of sets

• 

  Functional Sets, Part 4: Recursive Membership and Size · Brian Hicks View original

  Is this image relevant?

• 

  math mode - What is the best practice for typesetting set-builder notation with a large number ... View original

  Is this image relevant?

• 

  Set notation - Dave Tang's blog View original

  Is this image relevant?

• 

  Functional Sets, Part 4: Recursive Membership and Size · Brian Hicks View original

  Is this image relevant?

• 

  math mode - What is the best practice for typesetting set-builder notation with a large number ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Components of sets

• 

  Functional Sets, Part 4: Recursive Membership and Size · Brian Hicks View original

  Is this image relevant?

• 

  math mode - What is the best practice for typesetting set-builder notation with a large number ... View original

  Is this image relevant?

• 

  Set notation - Dave Tang's blog View original

  Is this image relevant?

• 

  Functional Sets, Part 4: Recursive Membership and Size · Brian Hicks View original

  Is this image relevant?

• 

  math mode - What is the best practice for typesetting set-builder notation with a large number ... View original

  Is this image relevant?

1 of 3

• A set is a well-defined collection of distinct objects
  • Objects in a set are called elements or members
• Sets can be represented using the roster method or set-builder notation
  • Roster method lists all elements within curly braces {} (1, 2, 3, 4, 5)
  • Set-builder notation describes the properties of elements using a variable and a condition {x | x is a positive integer less than 6}
• Sets are typically denoted using uppercase letters, while elements are represented by lowercase letters
• The symbol $\in$ denotes membership in a set
  • If A = {1, 2, 3}, then 1 $\in$ A (read as "1 is an element of set A")
• The empty set, denoted by {} or $\emptyset$, contains no elements
  • Empty set is a subset of every set, including itself
• The universal set, denoted by U, contains all possible elements under consideration for a given context

Finite vs infinite sets

• Sets can be classified as finite or infinite based on the number of elements they contain
  • Finite sets have a specific number of elements (a, b, c, d)
  • Infinite sets have an endless number of elements (1, 2, 3, 4, ...)
• The number of elements in a set is called its cardinality, denoted by |A| for a set A
  • For finite sets, the cardinality is the total number of elements
    • If A = {1, 2, 3, 4}, then |A| = 4
  • For infinite sets, the cardinality is denoted by $\aleph_0$ (aleph-null) or $\infty$ (infinity)
    • If B = {1, 2, 3, 4, ...}, then |B| = $\aleph_0$ or |B| = $\infty$
• Countable sets have a one-to-one correspondence with the set of natural numbers
  • Examples of countable sets: integers, rational numbers
• Uncountable sets have a cardinality greater than the set of natural numbers
  • Example of an uncountable set: real numbers

Equal vs equivalent sets

• Two sets are equal if they contain exactly the same elements
  • Order and repetition of elements do not matter in set equality
    • {1, 2, 3} = {3, 2, 1} = {1, 2, 2, 3}
  • Real-world example: A set of students in a class remains equal regardless of their seating arrangement or if their names are listed multiple times
• Two sets are equivalent if they have the same cardinality (number of elements)
  • Equivalent sets can have different elements as long as the number of elements is the same
    • {a, b, c} and {1, 2, 3} are equivalent sets because they both have 3 elements
  • Real-world example: A set of 20 apples and a set of 20 oranges are equivalent sets, even though they contain different types of fruit
• Equal sets are always equivalent, but equivalent sets may not be equal
  • {1, 2, 3} and {a, b, c} are equivalent but not equal

Set Operations and Relationships

• Union of two sets A and B, denoted by A ∪ B, is the set of all elements that belong to either A or B or both
• Intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that belong to both A and B
• The power set of a set A, denoted by P(A), is the set of all possible subsets of A, including the empty set and A itself
• Venn diagrams are visual representations used to illustrate relationships between sets, showing overlaps and distinctions between different sets