Mathematics
Binary operations
द्विआधारी संक्रियाएँ
In Class 12 Mathematics, this topic introduces binary operations as functions that combine two elements of a set to produce another element of the same set. Within the chapter Relations and Functions, students learn to identify whether an operation is well-defined and closed, represent it through tables or algebraic rules, and examine key properties such as commutativity, associativity, identity elements, and inverses. Examples involving familiar number systems help connect these ideas with broader function concepts.
Practice questions
On (\mathbb{R}), (a*b=a+b-3). What is the inverse of (8)?On (\mathbb{R}), (a*b=a+b-3). What is the value of ((2*5)*7)?On (\mathbb{R}), (a*b=a+b-3). What is the simplified form of (a*(b*c))?If (a*b=a+b+\lambda ab) and (\lambda\neq0), what is the identity element?On (\mathbb{R}), (a*b=a+b+\lambda ab). Under which condition will the inverse of (a) exist?In (a*b=a+b+\lambda ab) on (\mathbb{R}), (\lambda=2). What is the inverse of (4)?On (A={1,2,4,8}), (a*b=\operatorname{lcm}(a,b)). Which statement is correct?On the set (A=\mathbb{R}\setminus{-1}), the operation (a*b=a+b+ab) is defined. What is the identity element?On (A=\mathbb{R}\setminus{-1}), (a*b=a+b+ab). What is the inverse of (a)?On (\mathbb{Z}), (a*b=a+b-3) is defined. Which statement is correct?On (\mathbb{Z}), (a*b=a+b-3). What is the inverse of (a) under this operation?On positive real numbers, (a*b=\sqrt{ab}). Choose the correct conclusion about this operation.On (A={0,1,2,3}), (a*b) is the remainder when (a+b) is divided by (4). What is the inverse of (2)?On (A={1,2,3,4}), (a*b=\max(a,b)). Which is the identity element?On (A={1,2,3,4}), (a*b=\min(a,b)). What is the identity element?On non-zero real numbers, (a*b=\frac{ab}{2}). What is the identity element?On non-zero real numbers, (a*b=\frac{ab}{2}). Which is the inverse of (a)?On (\mathbb{R}), (a*b=a+b+1). What is the inverse of (a) under this operation?On (\mathbb{R}), (a*b=a+b+ab+2). Does this operation have an identity element?On (\mathbb{N}), (a*b=a^b). Which statement is correct?