What is the set of positive even integers, with the usual addition form?

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A finite group

Only a semi-group

Only a monoid

An infinite group

answer is B.

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Detailed Solution

Concept- By examining which operations the given set and binary operation follows, we will first understand the properties that must be available for the set and binary operation to be successful in obtaining all of the previously described possibilities.
Now let us define a group, monoid, and semigroup first:
Group theory is the study of groups, which are systems composed of a set of elements and a binary operation that may be applied to two elements of the set that satisfy particular axioms. Groups are used in modern algebra. A group X must apply to certain axioms if it has the binary operation *. Axioms are including:
The group has to be closed for the required binary operation, which

.
It must refer to associative law, which is

.
It must provide a unique identity that is

.
It must have an inverse that is

.
In this case, b is the inverse of a.
Let us consider the set of Integers and binary operation ‘+’.
Since the sum of two positive integers will always be a positive integer, if m,n

ϵ

Z+, then m+n

ϵ

Z+. Therefore, it is closed under "+".
If m,n,p

ϵ

Z+, then

. As a result, it is associative.
If m

ϵ

Z+, then

, but that is only possible if

but

. Therefore, there is no identity.
In this case, even the third property is not fulfilled; therefore we don't even need to examine the fourth.
We have a semi-group in this situation per the definition of a semi-group.
As a result, it is a semi-group.
Hence, the correct answer is option 2).

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