RD Sharma Class 12 Solutions Chapter 3 - Binary Operations (Ex 3.5) Exercise 3.5 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 3 - Binary Operations (Ex 3.5) Exercise 3.5
1. How do I use the RD Sharma Class 12 Solutions for Exercise 3.5 to find the identity element for a binary operation from a composition table?
To find the identity element, denoted by 'e', from a composition table, you should follow this method as shown in the solutions. Look for a row in the table that is identical to the top header row. The element at the beginning of this row is the identity element. Similarly, find a column that is identical to the left-most header column; the element at the top of this column will be the same identity element 'e'. For an operation * on set A, 'e' must satisfy the condition a * e = e * a = a for all 'a' in A.
2. What is the step-by-step method to determine the inverse of an element using a composition table in Exercise 3.5?
The solutions for Exercise 3.5 demonstrate this process clearly. First, you must identify the identity element (e) of the operation. To find the inverse of a specific element 'a', follow these steps:
Locate the row corresponding to element 'a'.
Move across this row until you find the identity element 'e'.
The element at the top of the column where 'e' is found is the inverse of 'a'.
The inverse of 'a', denoted as a⁻¹, must satisfy the condition a * a⁻¹ = a⁻¹ * a = e.
3. How can I quickly verify if a binary operation in Exercise 3.5 is commutative just by looking at its composition table?
A binary operation is commutative if a * b = b * a for all elements a, b. You can verify this property from a composition table by checking its symmetry. If the table is symmetric with respect to its main diagonal (the line of elements from the top-left to the bottom-right corner), then the operation is commutative. This means the entry in the i-th row and j-th column is the same as the entry in the j-th row and i-th column.
4. While Binary Operations are de-emphasised in the CBSE/NCERT syllabus for 2025-26, why is it still important to solve RD Sharma Chapter 3, Exercise 3.5?
Solving RD Sharma Chapter 3, especially Exercise 3.5, is crucial for building a strong mathematical foundation, even if the topic has reduced weightage in the CBSE board exam. These concepts are fundamental for higher-level mathematics and are frequently tested in competitive entrance exams like JEE Main and Advanced. Mastering binary operations, identity, and inverse elements helps develop abstract reasoning and problem-solving skills that are applicable across various mathematical fields.
5. Why is it necessary to find the identity element before finding the inverse of an element in a binary operation?
The concept of an inverse is fundamentally linked to the concept of an identity element. The inverse of an element 'a' is defined as the element 'b' which, when operated with 'a', results in the identity element 'e' (i.e., a * b = e). Without knowing what the identity element 'e' is, you have no target value to look for in the composition table. Therefore, identifying 'e' is always the mandatory first step before you can begin to search for the inverse of any element.
6. What is a common mistake when finding the inverse of an element for a non-commutative binary operation?
A common mistake is assuming that if a * b = e, then b * a must also equal e. For a non-commutative operation, this is not guaranteed. The definition of an inverse requires that both a * b = e and b * a = e. When using a composition table for a non-commutative operation, after finding a potential inverse 'b' by checking the row for 'a', you must also cross-check in the row for 'b' to ensure that b * a also yields the identity element. If it doesn't, the inverse does not exist.
7. How does solving for identity and inverse in Exercise 3.5 differ when an operation is defined by a formula versus a composition table?
The underlying principle is the same, but the method differs:
With a composition table (like in Ex 3.5): You use a visual search method. You scan rows and columns to locate the identity element and then use that to find inverses.
With a formula (e.g., a * b = a + b - 2): You use an algebraic method. To find the identity 'e', you solve the equation a * e = a (which becomes a + e - 2 = a). To find the inverse a⁻¹, you solve the equation a * a⁻¹ = e using the value of 'e' you just found.
The table method is practical for finite sets, while the algebraic method is necessary for infinite sets.
8. What happens if an entire row in a composition table does not contain the identity element?
If the row corresponding to an element 'a' does not contain the identity element 'e' anywhere, it signifies that there is no element 'b' in the set such that a * b = e. Consequently, the element 'a' does not have a right inverse, and therefore, it does not have an inverse under that binary operation. For an element to have an inverse, the identity element must appear in both its corresponding row and column.
