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