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). What is the inverse of (3)?On real numbers, (a*b=a+b-ab). What is the inverse of (3)?On positive real numbers, (a*b=ab). What is the identity element of this operation?On positive real numbers, (a*b=ab). What is the inverse of (5)?On integers, (a*b=ab). Does (2) have an inverse in integers?For (a*b=a+b+ab) on real numbers, which element does not have an inverse?For (a*b=a+b-ab) on real numbers, which element does not have an inverse?On (A={1,2,4,8}), (a*b=\gcd(a,b)). What is the identity element?On (A={1,2,4,8}), (a*b=\operatorname{lcm}(a,b)). What is the identity element?If (a*b=2a+b), which statement about ((1*2)*3) and (1*(2*3)) is correct?On the set (A={1,2,3,4}), the operation (a*b=\min(a,b)) is defined. What is the identity element of this operation?On the set (A={1,2,3,4,5}), the operation (a*b=\max(a,b)) is defined. Which element is the identity element?On real numbers, (a*b=a+b+ab) is defined. What is the identity element of this operation?On real numbers, (a*b=a+b-2) is defined. What is the inverse of (3) under this operation?On integers, (a*b=a+b+1) is defined. Choose the correct statement about this operation.On real numbers, (a*b=a-b) is defined. Which property does this operation have?On positive real numbers, (a*b=\frac{ab}{2}) is defined. What is the identity element of this operation?On real numbers, (a*b=a^2+b^2) is defined. Is this operation commutative?On real numbers, (a*b=a+2b) is defined. When will this operation be commutative?On (A={0,1,2}), define (a*b) as the remainder when (a+b) is divided by (3). What is the identity element?