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Questions and Answers
Which property is NOT one of the group axioms?
Which property is NOT one of the group axioms?
The identity element in a group is unique.
The identity element in a group is unique.
True
What is a group?
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 __________.
A group is a set with a binary operation satisfying the properties of associativity, identity element, and __________.
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Match the following sets with their operations to determine if they form a group:
Match the following sets with their operations to determine if they form a group:
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Which of the following is TRUE about a group?
Which of the following is TRUE about a group?
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All operations within a group must be commutative.
All operations within a group must be commutative.
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What determines if a set with an operation is a group?
What determines if a set with an operation is a group?
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Which of the following statements about a binary operation is true?
Which of the following statements about a binary operation is true?
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A binary operation is always commutative.
A binary operation is always commutative.
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What is an identity element in the context of a binary operation?
What is an identity element in the context of a binary operation?
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A binary operation is __________ if and only if a * b = b * a for all a, b in A.
A binary operation is __________ if and only if a * b = b * a for all a, b in A.
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Match the following terms with their definitions:
Match the following terms with their definitions:
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What must be true for a binary operation to have a unique identity element?
What must be true for a binary operation to have a unique identity element?
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If an operation has a left inverse, it must also have a right inverse.
If an operation has a left inverse, it must also have a right inverse.
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Can a set with a binary operation have an identity element that is not unique?
Can a set with a binary operation have an identity element that is not unique?
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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.
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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.
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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.
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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.