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A non empty set A is termed as an algebraic structure ________
- a)with respect to binary operation *
- b)with respect to ternary operation ?
- c)with respect to binary operation +
- d)with respect to unary operation –
Correct answer is option 'A'. Can you explain this answer?
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A non empty set A is termed as an algebraic structure ________a)with r...
A non empty set A is called an algebraic structure w.r.t binary operation “*” if (a*b) belongs to S for all (a*b) belongs to S. Therefore “*” is closure operation on ‘A’.
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A non empty set A is termed as an algebraic structure ________a)with r...
Understanding Algebraic Structures
An algebraic structure is a fundamental concept in mathematics, characterized by a set equipped with one or more operations that fulfill specific properties.
Definition of an Algebraic Structure
- A non-empty set A is termed an algebraic structure **with respect to a binary operation** when it has an operation * that combines any two elements of A to produce another element in A.
Binary Operations
- A binary operation is a function that takes two elements from a set and produces another element from the same set.
- For example, in the case of addition (+) or multiplication (×) defined on the set of real numbers, both operations take two numbers and yield another number within the same set.
Properties of Algebraic Structures
- **Closure**: For all a, b in A, the result of the operation a * b is also in A.
- **Associativity**: For all a, b, c in A, (a * b) * c = a * (b * c).
- **Identity Element**: There exists an element e in A such that for every a in A, e * a = a * e = a.
- **Inverse Elements**: For every a in A, there exists an element b in A such that a * b = b * a = e.
Why Other Options Are Incorrect
- **Ternary Operations**: Involve three elements, which do not define a standard algebraic structure as commonly understood in algebra.
- **Unary Operations**: Involve only one element and do not form a full algebraic structure on their own.
- **Specific Operations Like +**: While "+" is a binary operation, the question is more general and covers any binary operation.
In conclusion, the correct answer is option 'A' because an algebraic structure fundamentally relies on the concept of a binary operation that defines how elements of the set interact with each other.
An algebraic structure is a fundamental concept in mathematics, characterized by a set equipped with one or more operations that fulfill specific properties.
Definition of an Algebraic Structure
- A non-empty set A is termed an algebraic structure **with respect to a binary operation** when it has an operation * that combines any two elements of A to produce another element in A.
Binary Operations
- A binary operation is a function that takes two elements from a set and produces another element from the same set.
- For example, in the case of addition (+) or multiplication (×) defined on the set of real numbers, both operations take two numbers and yield another number within the same set.
Properties of Algebraic Structures
- **Closure**: For all a, b in A, the result of the operation a * b is also in A.
- **Associativity**: For all a, b, c in A, (a * b) * c = a * (b * c).
- **Identity Element**: There exists an element e in A such that for every a in A, e * a = a * e = a.
- **Inverse Elements**: For every a in A, there exists an element b in A such that a * b = b * a = e.
Why Other Options Are Incorrect
- **Ternary Operations**: Involve three elements, which do not define a standard algebraic structure as commonly understood in algebra.
- **Unary Operations**: Involve only one element and do not form a full algebraic structure on their own.
- **Specific Operations Like +**: While "+" is a binary operation, the question is more general and covers any binary operation.
In conclusion, the correct answer is option 'A' because an algebraic structure fundamentally relies on the concept of a binary operation that defines how elements of the set interact with each other.
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A non empty set A is termed as an algebraic structure ________a)with respect to binary operation *b)with respect to ternary operation ?c)with respect to binary operation +d)with respect to unary operation –Correct answer is option 'A'. Can you explain this answer?
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