Modern Algebra With respect to addition show that the set of

Modern Algebra

With respect to addition show that the set of Even integers, 2 zt is a group but that the of Odd integers, 2z + 1, is not a group.

Solution

We know that a group G is an algebraic structure comprising a set of elements and equipped with an operation ‘. ‘ which combines any two elements to form a third element. Further:

Let Z₁ and Z₂ be the sets of all even and odd intgers respectively. It may be observed that, with respect to addition, Z₁ and Z₂ fulfil the axioms 2,3 and 4 ( addition is associative, 0 is the additive identity and –a is the additive inverse of a). However, while the sum of two even numbers is an even number, the sum of two odd numbers is not an odd number. Therefore, Z₁ is a group and Z₂ is not a group