Questions and Answers
What is the definition of an empty set?
A set with no elements.
Define isomorphism in the context of binary algebraic structures.
A function that is one-to-one and onto such that $orall x,y$ in $S$, $ ext{ϕ}(x * y) = ext{ϕ}(x) *' ext{ϕ}(y)$.
What is cardinality?
The number of elements in a set.
What does one-to-one mean in terms of a function?
Signup and view all the answers
What does it mean for a function to be onto?
Signup and view all the answers
What are disjoint sets?
Signup and view all the answers
What is a partition of a set?
Signup and view all the answers
What are cells of a partition?
Signup and view all the answers
What is an equivalence relation?
Signup and view all the answers
How does an equivalence relation relate to partitions?
Signup and view all the answers
What is an equivalence class?
Signup and view all the answers
What is a binary operation?
Signup and view all the answers
What does closure mean in an operation?
Signup and view all the answers
What defines a commutative operation?
Signup and view all the answers
What is an associative operation?
Signup and view all the answers
What is a group in abstract algebra?
Signup and view all the answers
What is an abelian group?
Signup and view all the answers
What is a subgroup?
Signup and view all the answers
What is the subgroup test?
Signup and view all the answers
What is a cyclic subgroup?
Signup and view all the answers
What is the generator of a cyclic subgroup?
Signup and view all the answers
What is the greatest common divisor?
Signup and view all the answers
What does it mean for two numbers to be relatively prime?
Signup and view all the answers
What is intersection in the context of sets?
Signup and view all the answers
What is a permutation of a set?
Signup and view all the answers
What is the symmetric group on n letters?
Signup and view all the answers
What does the theorem of Lagrange state?
Signup and view all the answers
Every group of prime order is ____________.
Signup and view all the answers
What does cyclic mean in abstract algebra?
Signup and view all the answers
What is a ring?
Signup and view all the answers
What are the properties a ring must have?
Signup and view all the answers
What is unity in a mathematical context?
Signup and view all the answers
What is a unit in a ring?
Signup and view all the answers
Study Notes
Set Theory Concepts
- Empty Set: A fundamental concept in set theory, defined as a set containing no elements.
- Cardinality: Represents the number of elements contained within a set.
- Disjoint Sets: Two sets are disjoint if they share no common elements.
- Intersection: Refers to elements that are present in both sets of a pair.
Functions and Relations
- Isomorphism: A bijective function between two algebraic structures that preserves operations.
- One-to-One Function: A function where different inputs lead to different outputs.
- Onto Function: A function where every element in the codomain is mapped by at least one element from the domain.
- Equivalence Relation: A relation that is reflexive, symmetric, and transitive, leading to partitioning of a set.
Partitions and Classes
- Partition: Decomposes a set into non-empty subsets, ensuring each element is assigned to exactly one subset.
- Cells of a Partition: The individual subsets formed as a result of partitioning.
- Equivalence Class: The subset of a set formed by an equivalence relation, grouping items that are equivalent.
Algebraic Operations
- Binary Operation: An operation that combines two elements from a set to produce another element within the same set.
- Closure: An operation is closed within a set if the result of the operation on any two elements of the set remains in the set.
- Commutative Operation: An operation where the order of elements does not affect the outcome (a * b = b * a).
- Associative Operation: An operation where grouping of elements does not affect the result ((ab)c = a(bc)).
Group Theory
- Group: A set equipped with an operation that satisfies closure, has an identity element, every element has an inverse, and the operation is associative.
- Abelian Group: A group in which the operation is commutative, allowing any two elements to be combined without regard to order.
- Subgroup: A subset of a group that is itself a group under the same operation.
- Subgroup Test: To confirm that a subset is a subgroup, it must contain the product and inverses of its elements.
Cyclic Groups
- Cyclic Subgroup: Generated by a single element, meaning every element in the subgroup is a power of that generator.
- Generator of a Cyclic Subgroup: The specific element in a group from which all elements of the subgroup can be derived.
Number Theory
- Greatest Common Divisor (GCD): The highest integer that can divide two numbers without leaving a remainder.
- Relatively Prime: Two integers are relatively prime if their GCD is 1.
Permutations and Groups
- Permutation of a Set: A bijective function rearranging elements of a set.
- Symmetric Group on n Letters (Sn): The group of all permutations of n symbols.
Theorems and Properties
- Theorem of Lagrange: In any finite group, the order of any subgroup divides the order of the group.
- Every Group of Prime Order: Such groups are cyclic, since they contain only a limited number of elements.
- Cyclic: A cyclic group is defined as one where all elements can be generated by a single element.
Ring Theory
- Ring: A set with two operations (commonly addition and multiplication) that satisfies certain properties.
- Properties of a Ring: Must form an Abelian group under addition, be associative under multiplication, and adhere to distributive laws.
- Unity: The multiplicative identity element in a ring denoted by 1.
- Unit: An element of a ring possessing a multiplicative inverse.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Explore essential concepts of set theory and functions in this quiz. Learn about empty sets, cardinality, functions like isomorphism, and the various types of relations. Test your understanding of partitions, disjoint sets, and more.