RD Sharma Class 12 Solutions Chapter 3 - Binary Operations (Ex 3.1) Exercise 3.1 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 3 - Binary Operations (Ex 3.1) Exercise 3.1
1. What is the correct method to verify if a given operation '*' is a binary operation on a set S in RD Sharma Exercise 3.1?
To verify if an operation '*' is a binary operation on a set S, you must prove that for any two elements a and b from S, the result of a * b is also an element of S. The step-by-step method is as follows:
- Select two arbitrary elements, 'a' and 'b', from the given set S.
- Apply the defined operation to get the result of a * b.
- Analyse the result to confirm if it will always belong to the set S, for all possible values of 'a' and 'b'. If it does, the operation is a binary operation on S.
2. How do you solve problems that require checking for commutativity and associativity of a binary operation in Class 12 Maths?
To solve problems checking these properties for an operation '*', you must perform two separate tests as per the CBSE pattern:
- For Commutativity: You must check if a * b = b * a. Calculate the expression for both sides separately. If the expressions are identical for all a, b in the set, the operation is commutative.
- For Associativity: You must check if (a * b) * c = a * (b * c). First, calculate the Left-Hand Side (LHS) by evaluating (a * b) and then applying the operation with 'c'. Next, calculate the Right-Hand Side (RHS) by evaluating (b * c) first. If LHS equals RHS, the operation is associative.
3. What are the precise steps to find the identity element for a binary operation?
Finding the identity element 'e' requires a systematic approach. Let '*' be the binary operation on a set S. The correct steps are:
- Assume an identity element 'e' exists within the set S.
- Use the definition of the identity element: a * e = a and e * a = a.
- Solve the equation 'a * e = a' for 'e'.
- The solution for 'e' must be a constant value that is independent of 'a'.
- Finally, verify that this value of 'e' is a member of the set S. If it is, then it is the identity element.
4. Why is finding the identity element a crucial first step before finding the inverse of an element?
The concept of an inverse is fundamentally dependent on the identity element. The definition of an inverse 'b' of an element 'a' is that their operation results in the identity element 'e' (i.e., a * b = e). Therefore, you cannot even begin to solve for the inverse 'b' without knowing what 'e' is. If a binary operation has no identity element, then none of the elements in the set can have an inverse.
5. What is a common mistake when determining the inverse of an element for a binary operation in Exercise 3.1?
A very common mistake is to assume the inverse is a standard mathematical reciprocal (like 1/x) or negative (-x). The inverse must be calculated specifically for the given binary operation. The correct procedure is to first identify the identity element 'e', and then solve the equation a * b = e for 'b', where 'b' is the inverse of 'a'. Another frequent error is forgetting to check if the calculated inverse 'b' actually belongs to the specified set. If it doesn't, then the element 'a' is not invertible within that set.
6. Can a binary operation be commutative but not associative? Explain with an example.
Yes, the properties of commutativity and associativity are independent. An operation can satisfy one without satisfying the other. For example, consider the operation a * b = (a + b) / 2 on the set of rational numbers, Q.
- It is commutative: a * b = (a+b)/2 and b * a = (b+a)/2. These are equal.
- It is not associative: (a * b) * c = ((a+b)/2) * c = ((a+b)/2 + c)/2 = (a+b+2c)/4. However, a * (b * c) = a * ((b+c)/2) = (a + (b+c)/2)/2 = (2a+b+c)/4. Since these results are not equal, the operation is not associative.
7. How do you prove that an identity element for any binary operation, if it exists, must be unique?
You can prove the uniqueness of an identity element using a proof by contradiction. Assume there are two different identity elements, e1 and e2, for an operation '*' on a set S.
- By the definition of an identity element, since e1 is an identity, it must be true that e1 * e2 = e2.
- Similarly, since e2 is an identity, it must also be true that e1 * e2 = e1.
- From these two statements, we can conclude that e1 = e2.
This contradicts our initial assumption that they were different. Therefore, if an identity element exists, it must be unique.
