by Albert Major

Introduction to Logic

Basics of Set Theory

Set theory is the study of collections of objects treated as single units, (like thinking about a box of items as one thing rather than individual pieces.)

• A set is simply a collection of things (called members) considered as one whole unit
  • Example: {Mira, Darth Vader} is a set with two members

Key notations:

• ∈ means "is a member of"
  • Example: Mira ∈ {Mira, Darth Vader} means Mira is in that set
• ∉ means "is not a member of"
  • Example: Nathan ∉ {Mira, Darth Vader} means Nathan isn't in that set
• {} or ∅ represents an empty set (a set with no members)

Axiom of extensionality

• Two sets are identical if and only if they have exactly the same members
• Two sets are distinct if and only if they have at least one distinct member

Important rules:

• Order doesn't matter in sets
  • {Socrates, Plato} = {Plato, Socrates}
• The empty set is a subset of every other set
• Two sets are identical if they have exactly the same members
• Sets can also stand in the membership relation to other sets
  • {Socrates, Batman} ∈ {{Socrates, Batman}, {Cicero, Superman}}
    • but be careful:
    • {Socrates} ∉ {{Socrates, Batman}, {Cicero, Superman}}

Subsets and supersets: