Is the set of positive integers a group under the operation of addition?

An integer is positive if it is greater than zero.

Answer: No. The set of positive integers is not a group under the operation of addition.

There are some rules under the operation of addition.

Explanation:

A group is a set with an operation under the addition that has the following 4 properties:

• The set is closed under the operation.
• The set is associative under the operation.
• The set has an identity element under the operation that is also an element of the set.
• Every element of the set has an inverse under the operation that is also an element of the set.

There is no identity element in the set of positive integers under the operation of addition. There is also at least one positive integer that does not have an inverse in the set of positive integers under the operation of addition.

The set {1,2,3,..........} under addition is not a group, because it does not satisfy all of the group properties as

• Does not have the identity element in the set. (o is the additive identity but not present in the set)
• Does not have the INVERSE PROPERTY because −1 and −2 are not present in the set.

Therefore, the set {1,2,3,..........} under addition is not a group.

Hence, the set of positive integers is not a group under the operation of addition.