RD Sharma Solutions for Chapter 3
FAQs on RD Sharma Class 12 Solutions Chapter 3 - Binary Operations (Ex 3.4) Exercise 3.4
1. How do you determine the identity element for a binary operation '*' on a set S, as required in RD Sharma Class 12 Ex 3.4?
To find the identity element (e) for a binary operation '*' on a set S, you must find an element that leaves any other element unchanged when the operation is applied. The step-by-step method is:
Assume 'e' is the identity element that belongs to the set S.
Apply the definition of an identity element: a * e = e * a = a for all 'a' in S.
Solve the equation 'a * e = a' to find a value for 'e'.
Verify that this value of 'e' is constant, independent of 'a', and belongs to the given set S. If it does, it is the identity element.
2. What is the correct method to find the inverse of an element 'a' for a binary operation in Chapter 3?
To find the inverse of an element 'a' (denoted as a⁻¹ or 'b'), you must first find the identity element 'e'. The inverse is the element that combines with 'a' to yield 'e'. The correct method is as follows:
Step 1: Calculate the identity element 'e' for the binary operation.
Step 2: Let 'b' be the inverse of 'a'.
Step 3: Set up the equation based on the definition of an inverse: a * b = e.
Step 4: Solve this equation for 'b' in terms of 'a'. The resulting expression for 'b' is the inverse of 'a'.
Step 5: Ensure that the calculated inverse 'b' exists within the specified set for the given 'a'.
3. How can you verify if a given operation is commutative but not associative using an example from Exercise 3.4?
To verify this, you need to test both properties. An operation '*' is commutative if a * b = b * a, and it is associative if (a * b) * c = a * (b * c). To prove it is one but not the other:
First, prove commutativity by showing that for any elements a and b from the set, the equation a * b = b * a holds true.
Next, to disprove associativity, find a counter-example. This means picking specific numerical values for a, b, and c from the given set and showing that (a * b) * c is not equal to a * (b * c). For instance, subtraction on integers is commutative for some cases but not generally, and it is not associative.
4. Why is it crucial to first find the identity element before calculating the inverse of an element in a binary operation?
It is crucial because the very definition of an inverse element is dependent on the identity element. The inverse of an element 'a' is another element 'b' such that when they are combined using the operation, the result is the identity 'e' (i.e., a * b = e). Without knowing the value of 'e', you cannot set up or solve the equation to find the inverse 'b'. The identity element acts as the target result for the inverse operation.
5. What are the steps to check if an operation '*' on a set, like Q (rational numbers), is a valid binary operation?
To check if an operation '*' is a valid binary operation on a set S, you must verify the closure property. This property states that for any two elements 'a' and 'b' from set S, the result of 'a * b' must also be an element of S. The steps are:
Take two arbitrary elements, 'a' and 'b', from the given set (e.g., Q).
Perform the operation 'a * b'.
Analyse the result to ensure it always falls within the set S for all possible choices of 'a' and 'b'. If you can find even one case where 'a * b' is not in S, the operation is not a binary operation on that set.
6. What is a common mistake when solving problems in Ex 3.4 involving binary operations on sets like R - {-1}?
A common mistake when dealing with a set like R - {-1} (the set of all real numbers except -1) is failing to check if the result of the operation can equal the excluded value. For an operation to be binary on this set, the result `a * b` can never be -1. Students often prove closure for all other real numbers but forget this critical step. For example, if `a * b = a + b + ab`, you must verify if `a + b + ab = -1` has any solutions where `a` and `b` are in `R - {-1}`. If it does, the operation is not binary on that set.
