Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > A group (M,*) is said to be abelian if ______...
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A group (M,*) is said to be abelian if ___________
- a)(x+y)=(y+x)
- b)(x*y)=(y*x)
- c)(x+y)=x
- d)(y*x)=(x+y)
Correct answer is option 'B'. Can you explain this answer?
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A group (M,*) is said to be abelian if ___________a)(x+y)=(y+x)b)(x*y)...
Abelian Group in Mathematics
An abelian group is a mathematical structure consisting of a set of elements and an operation that combines any two elements to form a third element, satisfying certain axioms. In an abelian group, the operation is commutative, meaning that the order of the elements does not matter.
Definition of an Abelian Group
A group (M, *) is said to be abelian if the operation * is commutative, i.e., for any two elements x and y in the group, the result of their operation is the same regardless of the order in which they are combined. Symbolically, this can be written as:
(x * y) = (y * x)
Explanation of the Correct Answer
The correct answer to the question is option B, which states that (x * y) = (y * x). Let's understand why this is the correct answer.
Commutative Property of Abelian Groups
The commutative property, also known as the commutativity or commutative law, states that the order of the elements does not affect the result of the operation. In the context of abelian groups, it means that for any two elements x and y in the group, the result of their operation is the same regardless of the order in which they are combined.
Example
Let's consider a specific example to illustrate the commutative property in an abelian group. Suppose we have a group of integers under addition, denoted as (Z, +). In this group, the operation is addition (+).
If we take two integers, x = 2 and y = 3, and perform the addition operation in the given order, we get:
(x + y) = (2 + 3) = 5
Now, if we change the order of the elements and perform the addition again, we get:
(y + x) = (3 + 2) = 5
As we can see, the result is the same, regardless of the order in which the elements are combined. This property holds true for all elements in the group, making it an abelian group.
Conclusion
In summary, an abelian group is a mathematical structure in which the operation is commutative, meaning that the order of the elements does not affect the result. The correct answer to the given question is option B, which correctly identifies the commutative property as the defining characteristic of an abelian group.
An abelian group is a mathematical structure consisting of a set of elements and an operation that combines any two elements to form a third element, satisfying certain axioms. In an abelian group, the operation is commutative, meaning that the order of the elements does not matter.
Definition of an Abelian Group
A group (M, *) is said to be abelian if the operation * is commutative, i.e., for any two elements x and y in the group, the result of their operation is the same regardless of the order in which they are combined. Symbolically, this can be written as:
(x * y) = (y * x)
Explanation of the Correct Answer
The correct answer to the question is option B, which states that (x * y) = (y * x). Let's understand why this is the correct answer.
Commutative Property of Abelian Groups
The commutative property, also known as the commutativity or commutative law, states that the order of the elements does not affect the result of the operation. In the context of abelian groups, it means that for any two elements x and y in the group, the result of their operation is the same regardless of the order in which they are combined.
Example
Let's consider a specific example to illustrate the commutative property in an abelian group. Suppose we have a group of integers under addition, denoted as (Z, +). In this group, the operation is addition (+).
If we take two integers, x = 2 and y = 3, and perform the addition operation in the given order, we get:
(x + y) = (2 + 3) = 5
Now, if we change the order of the elements and perform the addition again, we get:
(y + x) = (3 + 2) = 5
As we can see, the result is the same, regardless of the order in which the elements are combined. This property holds true for all elements in the group, making it an abelian group.
Conclusion
In summary, an abelian group is a mathematical structure in which the operation is commutative, meaning that the order of the elements does not affect the result. The correct answer to the given question is option B, which correctly identifies the commutative property as the defining characteristic of an abelian group.
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A group (M,*) is said to be abelian if ___________a)(x+y)=(y+x)b)(x*y)...
A group (M,*) is said to be abelian if (x*y) = (x*y) for all x, y belongs to M. Thus Commutative property should hold in a group.
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