# Set Theory Vocabulary Puzzles

Our new Set Theory vocabulary puzzles are a great way to hone students’ math vocabulary skills. This new version of our puzzles does NOT require any Java applets. We have crosswords puzzles with three levels of difficulty as well as a word search. All resources are interactive, engaging, and include a timer. Solutions are also provided. Choose a puzzle below to get started, and be sure to try our related activities!

## What Is Set Theory?

Set theory is a fundamental branch of mathematics that deals with the study of sets, which are collections of distinct objects called elements. These elements can be anything – numbers, letters, colors, or even other sets. Set theory provides a foundation for nearly all of mathematics and serves as a framework for defining mathematical structures, relationships, and operations.

### Basic Concepts

Set – A set is a well-defined collection of distinct objects. Sets are denoted by curly braces {}, and elements are listed within these braces, separated by commas. For example, A={1,2,3} represents a set containing the elements 1, 2, and 3.

Element – An element is an object that belongs to a set. We use the symbol ∈ to denote membership. For example, if x is an element of set
A, we write x ∈ A.

Subset – A set B is considered a subset of set A if every element of B is also an element of A. We write this as B ⊆ A. If there exists at least one element in A that is not in B, then B is a proper subset of A, denoted as B ⊂ A.

Universal Set – The universal set, denoted by U, is the set containing all objects under consideration within a given context.

### Operations

Union (∪) – The union of two sets A and B, denoted as A ∪ B, is the set containing all elements that are in A, in B, or in both.

Intersection (∩) – The intersection of sets A and B, denoted as A ∩ B, is the set containing all elements that are in both A and B.

Complement (A’) – The complement of a set A with respect to the universal set U is the set of all elements in U that are not in A.

Difference (-) – The difference of sets A and B, denoted as A-B, is the set containing all elements that are in A but not in B.

### Set Theory Axioms

Set theory relies on a set of axioms, usually the Zermelo-Fraenkel (ZF) axioms, which provide the foundational rules for set manipulation. These axioms include the axiom of extensionality, axiom of pairing, axiom of union, axiom of power set, axiom of infinity, axiom of replacement, and axiom of choice.

Applications

Set theory forms the basis for various branches of mathematics, including:

Number theory – Sets of numbers such as integers, rational numbers, and real numbers are fundamental in number theory.

Topology – Concepts of open sets, closed sets, and neighborhoods are central to topology, which studies properties of space that are preserved under continuous deformations.

Abstract algebra – Sets equipped with algebraic operations, such as groups, rings, and fields, are studied in abstract algebra.

Logic – Set theory provides the foundation for formal logic and the study of mathematical proofs.

## Common Set Theory Vocabulary Words

Cardinality – The cardinality of a set refers to the number of elements it contains. It provides a measure of the “size” of a set, whether finite or infinite.

Cartesian Product – The Cartesian product of two sets A and B is the set of all possible ordered pairs (a,b) where a is an element of A and
b is an element of B. It is denoted as A x B.

Complement – The complement of a set A with respect to a universal set U consists of all elements in U that are not in A. It is denoted as A’.

Countable Set – A countable set is one that can be put into a one-to-one correspondence with the natural numbers or a subset of the natural numbers. This means that its elements can be listed in a sequence.

Difference – The difference between two sets A and B is the set containing all elements that are in A but not in B. It is denoted as A – B.

Disjoint Sets – Disjoint sets are sets that have no elements in common. In other words, their intersection is the empty set.

Domain – The domain of a function is the set of all possible inputs for that function. It is the set of values for which the function is defined.

Empty Set – Also known as the null set, the empty set is the set containing no elements. It is denoted by ∅ or {}.

Element – An element is an object that belongs to a set. If x is an element of set A, it is written as x ∈ A.

Finite Set – A finite set is a set with a limited number of elements. The cardinality of a finite set is a natural number.

Function – A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. It assigns to each element of the domain exactly one element of the codomain.

Image – The image of a function is the set of all outputs produced by that function. It consists of all elements in the codomain that are the result of applying the function to elements in the domain.

Infinite Set – An infinite set is a set with an unlimited number of elements. Its cardinality is greater than that of any finite set.

Injective Function – An injective function, also known as a one-to-one function, is a function where distinct inputs map to distinct outputs. No two distinct elements in the domain map to the same element in the codomain.

Intersection – The intersection of two or more sets is the set containing only the elements that are common to all of those sets. It is denoted as A ∩ B.

Ordered Pair – An ordered pair is a pair of elements listed in a specific order. It is commonly used to represent relations or functions, with the first element being the input and the second being the output.

Partition – A partition of a set is a collection of non-empty, disjoint subsets whose union equals the original set. Each element of the original set belongs to exactly one subset in the partition.

Power Set – The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself. It is denoted as P(A) or 2 <sup>A</sup>.

Preimage – The preimage of an element in the codomain of a function is the set of all elements in the domain that map to that element. It is also known as the inverse image or the inverse image set.

Proper Subset – A proper subset of a set is a subset that is not equal to the original set. In other words, it contains some but not all of the elements of the original set.

Relation – A relation is a set of ordered pairs, where the first element in each pair is related to the second element. Relations can represent connections or associations between elements of sets.

Subset – A subset of a set is a set that contains only elements that are also in another set. It may include all elements of the original set.

Surjective Function – A surjective function, also known as an onto function, is a function where every element in the codomain is mapped to by at least one element in the domain. The function covers the entire codomain.

Tuple – A tuple is an ordered list of elements. Tuples can have a finite number of elements and are commonly used in mathematics and computer science to represent structured data.

Union – The union of two or more sets is the set containing all elements that are in any of those sets. It is denoted as A ∪ B.

Uncountable Set – An uncountable set is a set that cannot be put into a one-to-one correspondence with the natural numbers or any subset of the natural numbers. It has a cardinality greater than that of any countable set.

Universal Set – The universal set is the set that contains all objects under consideration within a given context. It serves as a reference point for defining subsets and performing set operations.

Variable – In set theory, a variable often represents an unspecified element or elements of a set. It is used in statements about sets, functions, and relations.

Zermelo-Fraenkel (ZF) Axioms – The Zermelo-Fraenkel axioms are a set of axioms that provide the foundational rules for set theory. They include axioms related to sets, functions, and the existence of certain sets.

Zero Element – The zero element of a set is an element that acts as an identity with respect to a specific operation defined on that set. It may also refer to the neutral element of the set under certain operations.