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+b-ab). Which statement is correct?On (\mathbb{R}\setminus{0}), (a*b=\frac{ab}{a+b}), where (a+b\neq0). Why is it not a binary operation on the whole set?On (A=\mathbb{R}\setminus{-1}), (a*b=a+b+ab). What is the inverse of (a) under this operation?On (A=\mathbb{R}\setminus{-1}), (a*b=a+b+ab). Why is this operation closed in (A)?On (A=\mathbb{R}\setminus{0}), (a*b=ab). Which statement is correct for this operation?On (A=\mathbb{Q}\setminus{0}), (a*b=\frac{a}{b}). Is it a binary operation?On (\mathbb{Q}\setminus{0}), (a*b=\frac{a}{b}). What kind of operation is it?On integers, (a*b=a+b-1). Which will be the identity element?On integers, (a*b=a+b-1). Which is the inverse of (a)?On (A={0,1,2}), (a*b) is defined as the remainder when (a+b) is divided by (3). What is the identity element?On (A={0,1,2}), (a*b) is the remainder when (a+b) is divided by (3). What is the inverse of (2)?On (A={1,2,3,4}), (a*b) is the remainder when (ab) is divided by (5), and remainder (0) is not replaced by (5). Is this a binary operation on (A)?On (A={1,2,3,4}), (a*b) is the remainder when (ab) is divided by (5). Which is the identity element?On (A={1,2,3,4}), (a*b) is the remainder when (ab) is divided by (5). What is the inverse of (3)?On real numbers, (a*b=a+b+ab+1). Which simple form is connected to this operation?On real numbers, (a*b=ab+a+b+1). Does this operation have an identity element?On real numbers, (a*b=2a+2b). In which property does this operation fail?On real numbers, (a*b=ka+kb). For which (k) will this operation be associative?On real numbers, (a*b=a+b+k). What will be the inverse of (a) under this operation?On (A=\mathbb{R}\setminus{0}), (a*b=\frac{ab}{2}). Which statement is correct?