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 real numbers, (a*b=a^2+b^2). It is a binary operation, but which property generally fails?On real numbers, (a*b=|a-b|). In which property does this operation fail?On real numbers, (a*b=|a|b). Why is this operation not commutative?On (A={1,-1}), usual multiplication is taken as the operation. What is the inverse of (-1)?On (A={1,-1}), (a*b=ab). What kind of operation is it?On natural numbers, (a*b=a+b-2). When does this operation fail to be closed on natural numbers?On integers, (a*b=a+b-2). Which is the identity element?On integers, (a*b=a+b-2). What is the inverse of (5)?On (A=\mathbb{R}\setminus{1}), (a*b=a+b-ab). Why is this operation closed in (A)?On real numbers, (a*b=a+b-ab). Which element has no inverse under this operation if the set is the whole (\mathbb{R})?On real numbers, (a*b=a+b+\lambda ab). Which statement is correct, where (\lambda) is a fixed real number?On real numbers, (a*b=a+b+\lambda ab). What is the identity element of this operation?On real numbers, (a*b=a+b+\lambda ab). When will the inverse of (a) be undefined?On \(A=\mathbb{R}\setminus\left{-\frac{1}{\lambda}\right}\), \(a*b=a+b+\lambda ab\), where \(\lambda\neq0\). What is the inverse of (a)?On positive real numbers, (a*b=\sqrt{ab}) is defined. Does this operation have an identity element?On real numbers, (a*b=a+2b). Which property does this operation not satisfy?On (\mathbb{R}), (a*b=a+b-ab). What is the value of (2*(3*4))?If (A=\mathbb{R}\setminus{1}) has the operation (a*b=a+b-ab), what is the identity element for this operation?On the set (\mathbb{R}\setminus{-1}), (a*b=a+b+ab) is defined. What is the inverse of (a)?If (a*b=a+b+2ab) is defined on (\mathbb{Z}), why is it a binary operation on (\mathbb{Z})?