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