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Set Theory Review Guide
Quiz
Instructions: Answer each question in 2-3 sentences.
- Define a set and give an example of roster notation.
- What does it mean for set elements to be unique and unordered?
- Explain the concept of a subset and provide an example.
- What is the difference between the union and intersection of two sets?
- Describe the complement of a set. How does it relate to the universal set?
- What is the purpose of set builder notation? Give an example.
- Explain the concept of a Cartesian product and provide an example.
- How is the relative complement different from the absolute complement?
- What is the symmetric difference of two sets?
- Define a power set and explain how to determine its cardinality.
Quiz Answer Key
- A set is a collection of distinct objects. Roster notation lists the elements within curly braces, e.g., A = {1, 2, 3}.
- Unique elements means a set doesn’t include the same object multiple times. Unordered means the arrangement of elements doesn’t matter, so {a, b} is the same set as {b, a}.
- A subset (B) contains only elements also found in another set (A). All elements of B are in A, denoted B ⊆ A. For example, if A = {1, 2, 3}, then B = {1, 2} is a subset of A.
- The union of two sets (A ∪ B) includes all elements in either A or B. The intersection (A ∩ B) includes only elements present in both A and B.
- The complement of a set (A’) contains all elements within the universal set (U) that are not in A. The universal set encompasses all relevant elements for a given situation.
- Set builder notation defines sets based on a property. The format is {x | property of x}. For example, {x | x is an even number} represents the set of all even numbers.
- The Cartesian product (A × B) of sets A and B contains all possible ordered pairs where the first element comes from A and the second from B. For example, if A = {1, 2} and B = {a, b}, then A × B = {(1, a), (1, b), (2, a), (2, b)}.
- The relative complement (A – B) includes elements in A but not in B. The absolute complement (A’) is the relative complement with respect to the universal set, meaning it contains elements in the universal set but not A.
- The symmetric difference (A Δ B) contains elements that are in either A or B, but not in both.
- The power set (P(S)) of set S is the set of all possible subsets of S, including the empty set and S itself. Its cardinality is 2 raised to the power of the cardinality of S (|P(S)| = 2^|S|).
Essay Questions
- Discuss the relationship between set theory and logic, particularly the connections between set operations and logical operators.
- Explain how the axioms of set theory establish a foundation for working with sets. Discuss the significance of the Axiom of Foundation and its implications.
- Compare and contrast the concepts of ordinal and cardinal numbers. How are they defined, and what roles do they play in set theory?
- Describe the construction of the real numbers from the natural numbers within the framework of set theory. Explain the role of Dedekind cuts or Cauchy sequences in this process.
- Explore the implications of Russell’s Paradox for naive set theory. How do axiomatic set theories, such as Zermelo-Fraenkel set theory, address this paradox?
Glossary of Key Terms
- Set: A collection of distinct objects.
- Element: An object belonging to a set.
- Roster Notation: Listing set elements within curly braces.
- Set Builder Notation: Defining sets based on a property.
- Subset: A set whose elements are all contained within another set.
- Union: A set containing all elements from two or more sets.
- Intersection: A set containing elements common to two or more sets.
- Complement: A set containing elements in the universal set but not in a specified set.
- Universal Set: A set encompassing all relevant elements for a given situation.
- Empty Set: A set containing no elements.
- Cartesian Product: A set containing all possible ordered pairs with elements from two sets.
- Relative Complement: The set difference, elements in one set but not another.
- Absolute Complement: The complement relative to the universal set.
- Symmetric Difference: A set containing elements in either of two sets, but not in their intersection.
- Power Set: A set containing all possible subsets of a given set.
- Cardinality: The number of elements in a set.
- Ordinal Number: Represents a position in a sequence.
- Cardinal Number: Represents the size of a set.
- Dedekind Cut: A partition of the rational numbers used to define real numbers.
- Cauchy Sequence: A sequence of numbers that converges to a limit.
- Russell’s Paradox: A paradox in naive set theory, highlighting the issue of sets containing themselves.
- Axiomatic Set Theory: Formal systems with axioms to avoid paradoxes like Russell’s Paradox.
- Zermelo-Fraenkel Set Theory (ZF): An axiomatic set theory widely used as a foundation for mathematics.
- Axiom of Foundation: An axiom in ZF preventing infinite descending chains of set membership.
- Transitive Set: A set where elements of its elements are also elements of the set itself.
- Constructible Universe (L): A model of set theory where all sets are constructed from simpler sets, based on Gödel’s work.
- Elementary Submodel: A subset that reflects the truth of formulas in a larger model.
- Gödel Operations: Operations used to build sets in the constructible universe.
- Inaccessible Cardinal: A large cardinal with specific properties related to its size and closure under certain operations.
- Measurable Cardinal: An even larger cardinal that admits a non-trivial, two-valued measure.
