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Questions and Answers

Which property is NOT one of the group axioms?

Existence of an identity element

Existence of inverses

Commutativity (correct)

Associativity

The identity element in a group is unique.

True

What is a group?

A set equipped with a binary operation that satisfies the group axioms.

A group is a set with a binary operation satisfying the properties of associativity, identity element, and __________.

inverses Signup and view all the answers

Match the following sets with their operations to determine if they form a group:

Integers with addition = Group Integers with subtraction = Not a Group Rationals with addition = Group Integers with multiplication = Not a Group Signup and view all the answers

Which of the following is TRUE about a group?

Each element must have a unique inverse Signup and view all the answers

All operations within a group must be commutative.

False Signup and view all the answers

What determines if a set with an operation is a group?

It must satisfy the group axioms: associativity, existence of an identity element, and existence of inverses. Signup and view all the answers

Which of the following statements about a binary operation is true?

A binary operation is a type of function that applies to two elements of a set. Signup and view all the answers

A binary operation is always commutative.

False Signup and view all the answers

What is an identity element in the context of a binary operation?

An identity element is an element in a set that, when applied in a binary operation with any other element, results in that element. Signup and view all the answers

A binary operation is __________ if and only if a * b = b * a for all a, b in A.

commutative Signup and view all the answers

Match the following terms with their definitions:

Associative = Operation where (a * b) * c = a * (b * c) Identity Element = An element e such that a * e = a Inverse Element = An element that combines with another to yield the identity Commutative = Operation where a * b = b * a Signup and view all the answers

What must be true for a binary operation to have a unique identity element?

There must exist one element in the set that acts as the identity for all elements. Signup and view all the answers

If an operation has a left inverse, it must also have a right inverse.

False Signup and view all the answers

Can a set with a binary operation have an identity element that is not unique?

No Signup and view all the answers

Study Notes

Groups and Group Axioms

A group is a set with a binary operation satisfying the following properties:
- Associative: (a * b) * c = a * (b * c) for all a, b, c in the set.
- Identity Element: There exists an element e such that a * e = e * a = a for all a in the set.
- Inverse Element: For every element a in the set, there exists an element a⁻¹ such that a * a⁻¹ = a⁻¹ * a = e.

Examples of Groups

Integers with Addition: The set of integers with the operation of addition forms a group.
- The identity element is 0.
- The inverse of an integer a is -a.
Rational Numbers with Addition: The set of rational numbers with the operation of addition forms a group.
- The identity element is 0.
- The inverse of a rational number a is -a.
Real Numbers with Addition: The set of real numbers with the operation of addition forms a group.
- The identity element is 0.
- The inverse of a real number a is -a.

Non-Group Examples

Integers with Subtraction: The set of integers with the operation of subtraction does not form a group because it is not associative.
Integers with Multiplication: The set of integers with the operation of multiplication does not form a group because 0 does not have an inverse.
Rational Numbers with Multiplication: The set of rational numbers excluding 0 with the operation of multiplication forms a group.
- The identity element is 1.
- The inverse of a rational number a is 1/a.

Group Properties

Abelian Group: A group is Abelian if its binary operation is commutative (a * b = b * a).
Unique Identity Element: The identity element in a group is unique.
Unique Inverses: The inverses of an element in a group are unique.

Binary Operations

A binary operation on a set A is a rule that assigns an element of A to each pair of elements in A.
Associative: A binary operation is associative if and only if (a * b) * c = a * (b * c) for all a, b, c in A.
Commutative: A binary operation is commutative if and only if a * b = b * a for all a, b in A.
Identity Element: A binary operation has a neutral element if and only if there exists an element e in A such that a * e = e * a = a for all a in A.
Left Inverse: An element a in A has a left inverse if and only if there exists an element b in A such that b * a = e.
Right Inverse: An element a in A has a right inverse if and only if there exists an element c in A such that a * c = e.

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Description

Test your understanding of group theory and its axioms. This quiz covers the basic properties of groups, including examples such as integers, rational numbers, and real numbers under addition. Explore how these structures meet the defined criteria for being considered a group.