The binary operation * on set Q is
(i) defined as a * b = a – b.
For idenity element e, a * e = e * a = a.
But a * e = a – e # a and e * a = e – a # a.
∴ There is no identity element for this operation.
(ii) Binary operation * is defined as
a * b = a² + b²
Here, a * e = a² + e² # a
and e * a – e² + a² # a.
So, this operation * has no identity.
(iii) The binary operation * is defined as
a * b = a + ab
We have: a * e = a + ae # a
and e * a = e + ea # a.
∴ There is no identity element.
(iv) The binary operation * is defined as
a * b = (a – b)²
Put b = e. We get a * e = (a – e)² # a
and e * a = (e – a)² # a
for any value of e ∈ Q.
⇒ There is no identity element.
(v) The operation * is
a * b = \(\frac { ab }{ 4 }\).
and e * a = \(\frac { ab }{ 4 }\) # a
for any value of e ∈ Q.
∴ Operation * has no identity element.
(vi) The operation * is a * b = ab².
Put b = e. We get a * e = ae² # a
and e * a = ea² # a
for any value of e ∈Q.
⇒ There is no identity element.
Thus, these operations have no identity.
