Set Theory Syllabus

Set Theory syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Set Theory, not a summary of it.

1. 1

   Introduction to Sets

   4 topics

   • Definition of a Set
   • Notation and Representation of Sets
   • Types of Sets
   • Venn Diagrams
2. 2

   Operations on Sets

   5 topics

   • Union of Sets
   • Intersection of Sets
   • Difference of Sets
   • Complement of a Set
   • Cartesian Product of Sets
3. 3

   Properties of Sets

   8 topics

   • Commutative Properties
   • Associative Properties
   • Distributive Properties
   • De Morgan's Laws
   • Idempotent Laws
   • Identity Laws
   • Domination Laws
   • Absorption Laws
4. 4

   Relations and Functions

   7 topics

   • Definition of Relation
   • Types of Relations
   • Properties of Relations
   • Definition of Function
   • Types of Functions
   • Composition of Functions
   • Inverse of Functions
5. 5

   Advanced Topics in Set Theory

   7 topics

   • Cardinality of Sets
   • Finite and Infinite Sets
   • Countable and Uncountable Sets
   • Power Set
   • Zermelo-Fraenkel Set Theory
   • Axiom of Choice
   • Paradoxes in Set Theory

Set Theory flashcards

The complete Set Theory deck: 50 questions with worked answers.

1. What is the definition of a set?

   A set is a well-defined collection of distinct objects, called elements or members. "Well-defined" means it is objectively clear whether any given object belongs to the set or not.

2. What does it mean for a collection to be "well-defined" in set theory?

   It means there is a clear, unambiguous rule that determines whether any object is a member of the collection. For example, "the set of tall people" is not well-defined, but "the set of integers greater than $5$" is.

3. What symbols denote membership and non-membership in a set?

   $x \in A$ means "$x$ is an element of $A$", and $x \notin A$ means "$x$ is not an element of $A$".

4. Describe the roster (tabular) method of representing a set, with an example.

   The roster method lists all elements explicitly, separated by commas within braces. For example, the set of the first four positive even numbers is $A = \{2, 4, 6, 8\}$.

5. Describe the set-builder (rule) method of representing a set, with an example.

   Set-builder notation specifies elements by a defining property: $A = \{x \mid P(x)\}$, read "the set of all $x$ such that $P(x)$ holds". For example, $A = \{x \mid x \in \mathbb{N},\ x < 5\} = \{1, 2, 3, 4\}$.

6. In set notation, what do the symbols $\mid$ and $:$ mean inside set-builder form?

   Both are read as "such that", separating the variable from the property it must satisfy, as in $\{x \mid x > 0\}$ or $\{x : x > 0\}$.

7. What are the standard symbols for the sets of natural numbers, integers, rationals, reals, and complex numbers?

   $\mathbb{N}$ (naturals), $\mathbb{Z}$ (integers), $\mathbb{Q}$ (rationals), $\mathbb{R}$ (reals), and $\mathbb{C}$ (complex numbers).

8. What is the empty (null) set, and how is it denoted?

   The empty set is the set containing no elements. It is denoted $\varnothing$ or $\{\}$. Its cardinality is $0$.

9. What is a singleton set?

   A singleton set is a set containing exactly one element, e.g., $\{a\}$. Note that $\{a\}$ and $a$ are different: one is a set, the other is an element.

10. What is a finite set versus an infinite set?

    A finite set has a countable, limited number of elements (its cardinality is a non-negative integer). An infinite set has unlimited elements, e.g., $\mathbb{N} = \{1, 2, 3, \dots\}$.

11. What is the cardinality of a set, and how is it written?

    The cardinality is the number of elements in the set, written $|A|$ or $n(A)$. For $A = \{2, 4, 6\}$, $|A| = 3$.

12. What are equal sets?

    Two sets are equal if they contain exactly the same elements: $A = B$ means $x \in A \iff x \in B$. Order and repetition do not matter, so $\{1,2,3\} = \{3,2,1,1\}$.

13. What are equivalent sets?

    Two sets are equivalent if they have the same cardinality, i.e., $|A| = |B|$, even if their elements differ. For example, $\{a,b,c\}$ and $\{1,2,3\}$ are equivalent.

14. Define subset and proper subset.

    $A \subseteq B$ (subset) means every element of $A$ is in $B$. $A \subset B$ (proper subset) means $A \subseteq B$ and $A \neq B$ (so $B$ has at least one element not in $A$).

15. Is the empty set a subset of every set? Is a set a subset of itself?

    Yes to both: $\varnothing \subseteq A$ for every set $A$, and $A \subseteq A$ for every set $A$.

16. What is the power set of a set, and how many elements does it have?

    The power set $\mathcal{P}(A)$ is the set of all subsets of $A$. If $|A| = n$, then $|\mathcal{P}(A)| = 2^{n}$.

17. What is the universal set?

    The universal set $U$ is the set containing all objects under consideration in a particular context; every other set in that context is a subset of $U$.

18. What are disjoint sets?

    Two sets are disjoint if they have no elements in common, i.e., $A \cap B = \varnothing$.

19. What is a Venn diagram used for?

    A Venn diagram is a pictorial representation of sets using overlapping regions (usually circles) inside a rectangle (the universal set $U$), used to visualize relationships and operations among sets.

20. In a Venn diagram, how are the universal set and individual sets typically drawn?

    The universal set $U$ is drawn as a rectangle, and individual sets are drawn as circles (or closed curves) inside it. Overlapping regions represent common elements.

Set Theory syllabus FAQ