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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

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