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-ab). What is the identity element?On (A=\mathbb{R}\setminus{1}), (a*b=a+b-ab). Which statement is correct?On (\mathbb{Z}), (a*b=a+b+ab). Which option gives the best analysis for closure?On (A={x\in\mathbb{Z}:x\text{ is even}}), (a*b=\frac{a+b}{2}). Is it a binary operation?On (A={1,3,5,7}), (a*b) is the remainder when (a+b) is divided by (8). Is it closed on (A)?On (\mathbb{R}), (a*b=a^2+b^2). Why is this operation commutative?On (\mathbb{R}), (a*b=a^2+b^2). Is this operation associative?On (\mathbb{R}), (a*b=a+b+kab). For which values of (k) is the operation associative?On (\mathbb{R}\setminus{-\frac{1}{k}}), (a*b=a+b+kab), where (k\neq 0). What is the identity element?On (\mathbb{R}\setminus{-\frac{1}{2}}), (a*b=a+b+2ab). What is the inverse of (a)?On (\mathbb{R}), (a*b=a+2b). Choose the correct statement.On (\mathbb{R}), (a*b=2a+b). Does this operation have an identity element?On (\mathbb{R}), (a*b=a+b-ab). When will (a*b=1)?On (\mathbb{Q}\setminus{0}), (a*b=\frac{a}{b}). In which property does this operation fail?On (\mathbb{Q}\setminus{0}), (a*b=\frac{a}{b}). Is this operation associative?On (A={0,1}), the operation (a*b) is ordinary multiplication. Which element has no inverse?On (A={1,-1}), ordinary multiplication is the operation. What is the correct statement about this structure?On (\mathbb{R}), (a*b=a+b+ab). Which condition is necessary and sufficient for (a*b=b)?On (A=\mathbb{R}^{+}), (a*b=ab). In which set does the inverse of (a) lie?On (A=\mathbb{R}^{+}), (a*b=\frac{a+b}{2}). What is true about the identity element?