Set Theory Quiz and Flashcards for Effective Learning

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?

A function such that $f(x_1) = f(x_2)$ implies that $x_1 = x_2$. Signup and view all the answers

What does it mean for a function to be onto?

The range of the function is all of its codomain. Signup and view all the answers

What are disjoint sets?

Two sets that do not have any elements in common. Signup and view all the answers

What is a partition of a set?

A decomposition of S into non-empty subsets such that every element of S is in exactly one of the subsets. Signup and view all the answers

What are cells of a partition?

The subsets of a partition. Signup and view all the answers

What is an equivalence relation?

A relation that is reflexive, symmetric, and transitive. Signup and view all the answers

How does an equivalence relation relate to partitions?

An equivalence relation will partition a set. Signup and view all the answers

What is an equivalence class?

Each set created by an equivalence relation. Signup and view all the answers

What is a binary operation?

An operation on any two elements of a set that results in an element in the set. Signup and view all the answers

What does closure mean in an operation?

An operation is closed if for all elements $a,b$ in the set, $a * b$ is in the set. Signup and view all the answers

What defines a commutative operation?

An operation is commutative if $a * b = b * a$ for all elements in the set. Signup and view all the answers

What is an associative operation?

An operation is associative if $(a * b) * c = a * (b * c)$ for all elements in the set. Signup and view all the answers

What is a group in abstract algebra?

A set with an operation that is closed, has an identity, each element has an inverse, and is associative. Signup and view all the answers

What is an abelian group?

A group in which all the elements commute. Signup and view all the answers

What is a subgroup?

A subset of a group that is itself a group. Signup and view all the answers

What is the subgroup test?

A subset H of a group is a subgroup if for all a,b in H, $a * b^{-1}$ is in H. Signup and view all the answers

What is a cyclic subgroup?

A subgroup generated by a member of the group. Signup and view all the answers

What is the generator of a cyclic subgroup?

The element of the group that generates the subgroup. Signup and view all the answers

What is the greatest common divisor?

The largest factor that is common to two numbers. Signup and view all the answers

What does it mean for two numbers to be relatively prime?

If the greatest common divisor of two numbers is 1. Signup and view all the answers

What is intersection in the context of sets?

Elements that are common to both sets in a pair of two sets. Signup and view all the answers

What is a permutation of a set?

A function on a set that is both one-to-one and onto. Signup and view all the answers

What is the symmetric group on n letters?

Sn. Signup and view all the answers

What does the theorem of Lagrange state?

If G is a group and H is any subgroup of G, then the order of H must divide the order of G. Signup and view all the answers

Every group of prime order is ____________.

Cyclic. Signup and view all the answers

What does cyclic mean in abstract algebra?

The whole group can be generated by a single element. Signup and view all the answers

What is a ring?

A set with two operations, typically called + and *. Signup and view all the answers

What are the properties a ring must have?

It is an Abelian Group under +, associative under *, and satisfies the distributive laws. Signup and view all the answers

What is unity in a mathematical context?

The multiplicative identity, 1. Signup and view all the answers

What is a unit in a ring?

An element of a ring that has a multiplicative inverse. 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.

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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.

Set Theory and Functions Concepts