Page 20 - Relations and Functions 19.10.06.pmd

20 MATHEMATICS

and division are examples of binary operation, as ‘binary’ means two. If we want to
have a general definition which can cover all these four operations, then the set of
numbers is to be replaced by an arbitrary set X and then general binary operation is
nothing but association of any pair of elements a, b from X to another element of X.
This gives rise to a general definition as follows:
Definition 10 A binary operation ∗ on a set A is a function ∗ : A × A → A. We denote
∗ (a, b) by a ∗ b.
Example 29 Show that addition, subtraction and multiplication are binary operations
on R, but division is not a binary operation on R. Further, show that division is a binary
operation on the set R of nonzero real numbers.
∗ ∗ ∗ ∗ ∗
Solution +: R × R → R is given by
(a, b) → a + b
–: R × R → R is given by
(a, b) → a – b
×: R × R → R is given by
(a, b) → ab
Since ‘+’, ‘–’ and ‘×’ are functions, they are binary operations on R.
a
But ÷: R × R → R, given by (a, b) → , is not a function and hence not a binary
b

a
operation, as for b = 0, is not defined.
b
a
However, ÷ : R × R → R , given by (a, b) → is a function and hence a
∗
∗
∗
b
binary operation on R .
∗
Example 30 Show that subtraction and division are not binary operations on N.
Solution – : N × N → N, given by (a, b) → a – b, is not binary operation, as the image
of (3, 5) under ‘–’ is 3 – 5 = – 2 ∉ N. Similarly, ÷ : N × N → N, given by (a, b) → a ÷ b

3
is not a binary operation, as the image of (3, 5) under ÷ is 3 ÷ 5 = ∉ N.
5
Example 31 Show that ∗ : R × R → R given by (a, b) → a + 4b is a binary
2
operation.
Solution Since ∗ carries each pair (a, b) to a unique element a + 4b in R, ∗ is a binary
2
operation on R.