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-b|). Which statement is correct?On (\mathbb{R}), (a*b=|a-b|). Is there an identity element?On (\mathbb{N}), (a*b=\operatorname{lcm}(a,b)). What is the identity element?On (\mathbb{N}), (a*b=\gcd(a,b)). Does the identity element lie in (\mathbb{N})?On (A={1,2,3,6}), (a*b=\gcd(a,b)). What is the identity element?On (A={1,2,3,6}), (a*b=\operatorname{lcm}(a,b)). What is the identity element?On (A={0,1,2}), (a*b) is the remainder when (ab) is divided by (3). Which element has no inverse?On (A={1,2,4,5}), (a*b) is the remainder when (ab) is divided by (6). Which statement is correct?On (\mathbb{R}), (a*b=a+b-2ab). Which element has no inverse?On (A=\mathbb{R}\setminus{\frac{1}{2}}), (a*b=a+b-2ab). What is the inverse of (a)?On (\mathbb{R}), (a*b=a+b+\lambda). For which (\lambda) will the identity element be (5)?On (\mathbb{R}), (a*b=a+b+\lambda ab). For which (\lambda) will the element (2) have no inverse?On (\mathbb{R}), (a*b=a+b+mab). If the inverse of (2) is (-1), what is the value of (m)?On (A={0,1,2,3,4}), (a*b) is the remainder when (a+b) is divided by (5). What is (3*4)?On (A={0,1,2,3,4}), (a*b) is the remainder of (a+b) modulo (5). What is the inverse of (4)?If (A) has (n) elements, how many binary operations can be defined on (A)?If (A) has (n) elements, how many commutative binary operations can be defined on (A)?On (\mathbb{Z}), the operation (a*b=a+b+ab) is defined. Choose the correct statement.On (\mathbb{R}), the operation (a*b=a+b+4) is defined. What is its identity element?On (\mathbb{R}), (a*b=a+b+4). What is the inverse of (6)?