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}\setminus{0}), (a*b=\frac{a}{b}). Which statement is correct?On (\mathbb{R}^{+}), (a*b=\sqrt{ab}). What is the correct conclusion about this operation?On (\mathbb{R}^{+}), (a*b=\frac{ab}{a+b}). Which property does this operation satisfy?On (\mathbb{R}), (a*b=a+b-1). Which statement about ((2*3)*4) and (2*(3*4)) is correct?On (\mathbb{R}), (a*b=a+b-1). What is the value of ((2*3)*4)?On (\mathbb{R}), (a*b=a+b-1). What is the identity element of this operation?On (\mathbb{R}), (a*b=a+b-1). What is the inverse of (8)?On (\mathbb{R}), (a*b=a+b+ab). What is the solution of (x*2=8)?On \(\mathbb{R}\setminus{-2}\), \(a*b=a+b+\frac{ab}{2}\). What is the identity element of this operation?On (\mathbb{R}\setminus{-2}), (a*b=a+b+\frac{ab}{2}). What is the inverse of (4)?On (\mathbb{R}), (a*b=a^2+b). In which property does this operation fail?On (\mathbb{R}), (a*b=a+b+ab). Which form is most useful for explaining closure on (\mathbb{R}\setminus{-1})?On (\mathbb{R}), (a*b=a+b+ab). If (a*b=0), what is the correct value of (b) in terms of (a)?On (\mathbb{R}), (a*b=a+b-ab). If (a*b=0), what is the correct value of (b)?On (\mathbb{Z}), (a*b=a+b+ab). Which element is its own inverse?On (\mathbb{Z}), (a*b=a+b+ab). Which of the following elements is not its own inverse?On (S={1,2,3,4,6,12}), (a*b=\gcd(a,b)). What is the identity element of this operation?On (S={1,2,3,4,6,12}), (a*b=\operatorname{lcm}(a,b)). What is the identity element of this operation?On (\mathbb{R}^{+}), (a*b=\frac{a+b}{2}). Which statement is correct?On (\mathbb{R}), (a*b=a+b-ab). What is the solution of (2*x=3)?