Which of the following is an abelian group?Question 4Answera.Set of all integers with a relation a*b = ab for all a, b in the set.b.Set of odd integers with a relation a*b =  ab for all a, b in the set.c.Set of all integers with a relation a*b = 2ab for all a, b in the set.d.Set of odd integers with a relation a*b = 2ab for all a, b in the set.

An Abelian group is a set equipped with an associative binary operation that has an identity element and every element in the set has an inverse. The operation is also commutative, meaning that the order of the operation does not matter (ab = ba).

Let's analyze each option:

a. Set of all integers with a relation a*b = ab for all a, b in the set. This is not an Abelian group. The operation defined here is just the standard multiplication of integers, which does not have an identity element (there is no integer which can be multiplied with any integer to give that integer) and inverses do not exist for all elements (for example, the inverse of 2 under multiplication is 1/2, which is not an integer).

b. Set of odd integers with a relation a*b = ab for all a, b in the set. This is also not an Abelian group for the same reasons as option a.

c. Set of all integers with a relation ab = 2ab for all a, b in the set. This is not an Abelian group. The operation defined here is not associative (for example, (12)3 = 26 = 12, but 1*(23) = 16 = 6), it does not have an identity element and inverses do not exist for all elements.

d. Set of odd integers with a relation a*b = 2ab for all a, b in the set. This is also not an Abelian group for the same reasons as option c.

So, none of the options given is an Abelian group.