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{Q}\setminus{0}), (a*b=\frac{ab}{2}). What is the inverse of (5)?On (\mathbb{R}), (a*b=a+b+1). Which element acts as the identity that allows every real number to have an inverse?On (\mathbb{R}), (a*b=a+b-ab). Which element motivates restricting the operation to (\mathbb{R}\setminus{1})?On (\mathbb{R}\setminus{1}), (a*b=a+b-ab). What is the identity element?On (\mathbb{R}\setminus{1}), (a*b=a+b-ab). What is the inverse of (a)?On (\mathbb{Z}), (a*b=a+b+ab). Why does this operation not form a group on the whole set?On (\mathbb{R}), (a*b=a+b-ab). Which statement is correct?On (\mathbb{R}), (a*b=a+b-ab). What is the correct reason this operation is associative?On ({0,1}), (a*b=a+b-ab). This operation behaves like which common logical operation?On ({0,1}), (a*b=ab). Which statement is correct for this operation?On (\mathbb{R}), (a*b=a+b+ab) and (a\circ b=a+b). Under what condition does (*) distribute over (\circ)?On (\mathbb{R}), (a*b=a+b) and (a\circ b=ab). Which operation distributes over the other?On (\mathbb{R}), (a*b=a-b). Which of the following statements is correct?On (\mathbb{Z}), (a*b=a+b+3). What is the inverse of (4) under this operation?On (\mathbb{R}), (a*b=a+b+\lambda). For which (\lambda) will the identity be (5)?On (\mathbb{R}), (a*b=a+\mu b). The operation will be commutative only for what value of (\mu)?On (\mathbb{R}), (a*b=a+\mu b). Which values of (\mu) make this operation associative?On (\mathbb{R}), (a*b=pa+qb). What condition is needed for this operation to be commutative?On (\mathbb{R}), (a*b=pa+qb). Which condition is necessary and sufficient for associativity?On (\mathbb{R}), (a*b=a+b+\alpha ab). If the inverses of (2) and (3) are (-1) and (-\frac{3}{2}) respectively, what is (\alpha)?