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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?
- Existence of an identity element
- Existence of inverses
- Commutativity (correct)
- Associativity
The identity element in a group is unique.
The identity element in a group is unique.
True (A)
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 __________.
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:
Which of the following is TRUE about a group?
Which of the following is TRUE about a group?
All operations within a group must be commutative.
All operations within a group must be commutative.
What determines if a set with an operation is a group?
What determines if a set with an operation is a group?
Which of the following statements about a binary operation is true?
Which of the following statements about a binary operation is true?
A binary operation is always commutative.
A binary operation is always commutative.
What is an identity element in the context of a binary operation?
What is an identity element in the context of a binary operation?
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.
Match the following terms with their definitions:
Match the following terms with their definitions:
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?
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.
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.
- 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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