RD Sharma Class 12 Chapter 3 Free PDF
FAQs on RD Sharma Solutions for Class 12 Math Chapter 3 - Binary Operations - Free PDF Download
1. How do RD Sharma Solutions for Class 12 Chapter 3 help in mastering Binary Operations?
RD Sharma Solutions for Chapter 3 provide a comprehensive set of solved problems that go beyond the basic textbook exercises. They offer a step-by-step methodology for a wide range of questions, helping you understand how to verify properties like associativity, commutativity, and how to find identity and inverse elements. This helps build a strong foundation and tackle complex problems that often appear in competitive exams.
2. What defines an operation '*' as a binary operation on a set S?
An operation '*' is defined as a binary operation on a non-empty set S only if it satisfies the closure property. This means that for any two elements 'a' and 'b' taken from the set S, the result of the operation, 'a * b', must also be an element of the same set S. For example, addition is a binary operation on the set of natural numbers (N), but subtraction is not, because 3 - 5 = -2, which is not in N.
3. What are the main properties a binary operation can have?
The key properties used to classify and solve problems involving binary operations, as covered in RD Sharma, are:
Closure Property: For all a, b ∈ S, a * b ∈ S.
Associative Property: For all a, b, c ∈ S, (a * b) * c = a * (b * c).
Commutative Property: For all a, b ∈ S, a * b = b * a.
Existence of Identity: There exists an element e ∈ S such that for all a ∈ S, a * e = e * a = a.
Existence of Inverse: For each a ∈ S, there exists an element b ∈ S such that a * b = b * a = e, where 'e' is the identity element.
4. How can I determine the identity element for a binary operation on a set?
To find the identity element 'e' for a binary operation '*' on a set S, you must solve the equation a * e = a. The value you find for 'e' must be constant for all elements 'a' in the set S. Finally, you must verify that this value of 'e' is also an element of the set S. If it is not, then an identity element does not exist for that operation on that set.
5. What is the correct method to find the inverse of an element for a binary operation?
To find the inverse of an element 'a' in a set S, you must first confirm the existence of an identity element 'e'. Let the inverse of 'a' be 'b'. The correct method is to solve the equation a * b = e for 'b'. The resulting expression for 'b' will typically be in terms of 'a'. For an inverse to exist for a specific element 'a', its calculated inverse 'b' must also be a member of the set S.
6. How is the associative property fundamentally different from the commutative property?
The two properties address different aspects of an operation. The commutative property is about the order of elements (is a * b the same as b * a?). In contrast, the associative property is about the grouping of operations when there are three or more elements (is (a * b) * c the same as a * (b * c)?). An operation can be associative but not commutative, like matrix multiplication.
7. Why is the closure property the first thing to check when verifying a binary operation?
The closure property is the most fundamental requirement. If an operation on two elements from a set produces a result that falls outside that set, it fails the basic definition of a binary operation for that set. If an operation is not closed, it is not a binary operation, and therefore, checking for other properties like associativity or the existence of an identity element becomes irrelevant.
8. Is the 'Binary Operations' chapter part of the CBSE Class 12 Maths syllabus for the 2025-26 board exams?
No, as per the latest CBSE guidelines for the 2025-26 academic session, the chapter on Binary Operations has been removed from the Class 12 Maths board exam syllabus. However, the concepts of abstract algebra taught in this chapter are highly valuable for developing logical reasoning and form a foundational topic for various competitive engineering entrance exams and higher studies in mathematics.
