Associative Binary operation
Identity Binary Operation
Invertible Binary operation
Associative Binary operation
Binary operation ∗: A × A → A is said to be associative if( a ∗ b) ∗ c = a ∗( b ∗ c), ∀ a, b, c, ∈ A.
Numerical: Show that ∗: R × R → R given by a ∗ b → a + 2b is not associative.
Solution: For this function to be associative,( a ∗ b) ∗ c = a ∗( b ∗ c), LHS =( a ∗ b) ∗ c =( a + 2b) * c = a + 2b + 2c RHS = a ∗( b ∗ c) = a *( b + 2c) = a + 2b + 4c Since LHS ≠ RSH, so associative property is not true in this case
Identity Binary Operation
Given a binary operation ∗: A × A → A, an element e ∈ A, if it exists, is called identity for the operation ∗, if a ∗ e = a = e ∗ a, ∀ a ∈ A.
Numerical: Show that zero is the identity for addition on R and 1 is the identity for multiplication on R. But there is no identity element for the operations –: R × R → R and ÷: R∗ × R∗ → R∗.
Solution: 0 + a = a + 0 = a, so 0 is identity for addition operation in R 1 * a = a * 1 = a, so 1 is the identity for multiplication operation in R
There is no element e in R, such that a-e = e-a = a, so there is no identity for subtraction operation in R
Similarly, There is no element e in R, such that a / e = e / a = a, so there is no identity for division operation in R
Note: Zero is identity for the addition operation on R but it is not identity for the addition operation on N, as 0 ∉ N. In fact the addition operation on N does not have any identity.
Invertible Binary operation
Given a binary operation ∗: A × A → A with the identity element e in A, an element a ∈ A is said to be invertible with respect to the operation ∗, if there