Binary operations
Commutative binary operation
Notice that ( gof ) – 1 = f – 1 o g – 1
Binary operations
Addition , multiplication , subtraction and division are examples of binary operation , as ‘ binary ’ means two .
General binary operation is nothing but association of any pair of elements a , b from X to another element of X .
A binary operation ∗ on a set A is a function ∗ : A × A → A . We denote ∗ ( a , b ) by a ∗ b .
Numerical : Show that subtraction and division are not binary operations on N , while addition & multiplication are binary operation on N . N * N à N
Solution :
Case 1 : Addition . let ’ s see if N + N à N . Addition of two natural numbers gives natural number . So this is true .
Case 2 : Subtraction , let ’ s see if N -N à N . Subtraction of two natural numbers need not be a natural number . Eg : 1 -5 = -4 , which is not a Natural number .
Case 3 : Multiplication . let ’ s see if N X N à N . Multiplication of two natural numbers gives natural number . So this is true .
Case 4 : Division , let ’ s see if N / N à N . Division of two natural numbers need not be a natural number . Eg : 2 / 5 = 2 / 5 , which is not a Natural number .
Commutative binary operation
A binary operation ∗ on the set X is called commutative , if a ∗ b = b ∗ a , for every a , b ∈ X
Numerical : Show that ∗ : R × R→R defined by a ∗ b = a + 2b is not commutative . Solution : For commutative property to hold true a ∗ b = b ∗ a LHS = a ∗ b = a + 2b
ROHS = b * a = b + 2a Since LHS ≠ RSH , so commutative property is not true in this case