In RD Sharma Solutions for Class 12 Maths Chapter 3 binary operations, students begin to study the meaning and definition of the binary operation. The binary operation can be described as the set of numbers on which the binary operations are performed, and these operations include addition, subtraction, multiplication, division etc. These can be generalised as a binary operation performed in two elements, namely a and b from the set X. The final result of the operation on a and b is considered another element from the same set X.
Then, they move on to another topic called the number of binary operations, where they study how to define the number using some methods. Moreover, they also learn the different types of binary operations in an in-depth manner. The types of binary operations include Commutativity, Associativity and Distributivity.
The commutative property is defined as the property where the answer is unaffected by the change in the order of operation. It is a commutative operation if a * b = b * a, for all (a, b) A, which is a non-empty set. Let us consider addition as the operating binary operation for a = 9 and b = 5, so a + b = 14 = b + a.
Next, the associative property of binary operations hold if, for a non-empty set A, we can say (a * b) * c = a * (b * c). Incase N be termed as set of natural numbers and multiplication be the binary operation, let a = 2, b = 5, c = 4, we can write (a * b) * c = 40 = a * (b * c).
In distributive property, let ‘*’ and ‘o’ be two binary operations defined on a non-empty set A. The binary operations are considered to be distributive, only if a * (b o c) = (a * b) o (a * c) or (b o c) * a = (b * a) o (c * a).
Furthermore, students are also taught topics such as identity, the inverse of an element, the Composition table, and the Multiplication module.