Page 23 - Relations and Functions 19.10.06.pmd

RELATIONS AND FUNCTIONS 23

For the binary operation ‘+’ on R, the interesting feature of the number zero is that
a + 0 = a = 0 + a, i.e., any number remains unaltered by adding zero. But in case of
multiplication, the number 1 plays this role, as a × 1 = a = 1 × a, ∀ a in R. This leads
to the following definition:
Definition 13 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.

Example 38 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 a + 0 = 0 + a = a and a × 1 = a = 1 × a, ∀ a ∈ R implies that 0 and 1 are
identity elements for the operations ‘+’ and ‘×’ respectively. Further, there is no element
e in R with a – e = e – a, a. Similarly, we can not find any element e in R such that
∀
∗
a ÷ e = e ÷ a, ∀ a in R . Hence, ‘–’ and ‘÷’ do not have identity element.
∗
Remark 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.
One further notices that for the addition operation + : R × R → R, given any
a ∈ R, there exists – a in R such that a + (– a) = 0 (identity for ‘+’) = (– a) + a.

1
Similarly, for the multiplication operation on R, given any a ≠ 0 in R, we can choose
a
1 1
in R such that a × = 1(identity for ‘×’) = × a. This leads to the following definition:
a a
Definition 14 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 exists
an element b in A such that a ∗ b = e = b ∗ a and b is called the inverse of a and is
denoted by a .
–1
Example 39 Show that – a is the inverse of a for the addition operation ‘+’ on R and
1
a is the inverse of a ≠ 0 for the multiplication operation ‘×’ on R.

Solution As a + (– a) = a – a = 0 and (– a) + a = 0, – a is the inverse of a for addition.
1 1 1
Similarly, for a ≠ 0, a × = 1 = × a implies that is the inverse of a for multiplication.
a a a