Identity: A composition 
in a set
is said to possesses of an identity if there exists an element 
such that
Moreover, the element e, if it exists is called an identity element and the algebraic structure
is said to have an identity element with respect to
.
Examples: in a set is said to possesses of an identity if there exists an element such thatMoreover, the element e, if it exists is called an identity element and the algebraic structure
is said to have an identity element with respect to.
(1) If
, the set of real numbers then 
(Zero) is an additive identity of
because
the set of natural numbers, has no identity element with respect to addition because 
.
(2)
is the multiplicative identity of 
as
Evidently
is identity of multiplication for
(set of integers), 
(set of rational numbers, 
(set of real numbers).
, the set of real numbers then (Zero) is an additive identity of becausethe set of natural numbers, has no identity element with respect to addition because . (2)
is the multiplicative identity of asEvidently
is identity of multiplication for (set of integers), (set of rational numbers, (set of real numbers).
Inverse: An element 
is said to have its inverse with respect to certain operation
if there exists
such that
being the identity in
with respect to
.
is said to have its inverse with respect to certain operation if there exists such that being the identity in with respect to.
Such an element
, usually denoted by
is called the inverse of
. Thus 
for
.
, usually denoted by is called the inverse of. Thus for.
with respect to ordinary addition operation is
and in the set of non-zero rational numbers, the inverse of
with respect to multiplication is 
which belongs to the set. 
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