Let N 1 234ie set of natural numbers with the standard addi

Let N = {1, 2,3,4,...},i.e. set of natural numbers with the standard addition and scalar multiplication. Is N a real vector space. Justify your answer.

Solution

No N is not a vector space

As there does not exist the identity element e which is such that for any natural number \'n\'

n+e=e+n=n

Also there does not exist the additive inverse of any number.