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+2) and (a\circ b=a+b-2) are defined. What is the value of (3*(4\circ 5))?On (A={0,1}), (a*b) is the remainder when (a+b) is divided by (2). Which statement is correct for this operation?On positive integers, (a*b=\operatorname{lcm}(a,b)) is defined. What is the identity element of this operation?On positive integers, (a*b=\gcd(a,b)) is defined. Does this operation have an identity element?On real numbers, (a*b=a+b+ab) and (f(x)=x+1) are given. What is (f(a*b)) equal to?On real numbers, (a*b=a+b+ab) is defined. Evaluating (1*2*3) from left to right as ((1*2)*3), what is the value?On (A={1,2,3,4,5,6}), (a*b=\gcd(a,b)) is defined. What is the correct statement about the inverse of (4), if identity is assumed to be (6)?On (A={1,2,3,6}), (a*b=\operatorname{lcm}(a,b)) is defined. Which element is the inverse of (6), if the identity is (1)?On real numbers, (a*b=a+b+\lambda ab) is defined. For which (\lambda) will (0) be the identity element?On real numbers, (a*b=a+b+\lambda ab) is defined. If (\lambda\neq 0), which element will not have an inverse?On real numbers, (a*b=a+b-ab) is defined. Which element does not have an inverse?On (A={0,1,2}), (a*b) is the remainder when (a+b) is divided by (3). What is the value of (2*2)?On real numbers, (a*b=a+b-4ab) is defined. Choose the correct statement about (a=\frac{1}{4}).On real numbers, (a*b=a+b+ab) is defined. What is the solution of (x*2=7)?On real numbers, (a*b=2a+3b) is defined. Is this operation commutative?On real numbers, (a*b=2a+3b) is defined. Does this operation have an identity element?On real numbers, (a*b=a+b+m) is defined. If the identity element is (5), what is the value of (m)?On real numbers, (a*b=a+b+mab) is defined. For which (m) will the inverse of (2) be (-1)?On real numbers, (a*b=a+b+ab) is defined. If (a*b=0), how will (b) be written in terms of (a), where (a\neq -1)?On (A=\mathbb{R}\setminus{1}), (a*b=a+b-ab) is defined. Choose the correct statement about this operation on this set.