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^2+b^2) is defined. Which property does this operation satisfy?On (\mathbb{R}), (a*b=a-b) is defined. Choose the correct statement.If (a*b=2a+3b), what is the simplified form of (a*(b*c))?On (\mathbb{R}), (a*b=ka+b). For which (k) will the operation be associative?On (\mathbb{R}), (a*b=a+b+k). What is the identity element of this operation?On (\mathbb{R}), (a*b=a+b+5). What is the inverse of (a)?For (a*b=a+b-ab) on (\mathbb{R}), what is the value of (2*3)?On (\mathbb{R}), (a*b=a+b-ab). Which element is the inverse of (2)?On (A={0,1,2}), (a*b) is defined as the greater of (a) and (b). What is the identity element?On (A={0,1,2}), (a*b) is defined as the smaller of (a) and (b). What is the identity element?On (\mathbb{N}), (a*b=\gcd(a,b)) is defined. Does an identity element exist in (\mathbb{N})?On (\mathbb{N}), (a*b=\operatorname{lcm}(a,b)) is defined. What is the identity element of this operation?On (\mathbb{Z}), (a*b=a+b-1). Which property does this operation satisfy?On (\mathbb{Z}), (a*b=a+b-1). What is the inverse of (5)?On (\mathbb{Q}\setminus{0}), (a*b=\frac{ab}{2}). What is the identity element?On (\mathbb{Q}\setminus{0}), (a*b=\frac{ab}{2}). What will be the inverse of (a)?On (\mathbb{R}), (a*b=a+b+ab). On which set will this operation not be closed?For (a*b=a+b+ab) on (\mathbb{R}), which form is helpful for checking associativity?On (A=\mathbb{R}\setminus{0}), (a*b=\frac{a}{b}). Choose the correct statement.On (\mathbb{R}), (a*b=a+b+ab). Which statement about (a=-1) is correct?