I am new to linear algebra course. I am solving a problem in which a set $V$ is given which is not a vector space over field $F$ under the standard operation of addition and scalar multiplication. The problem is to identify which axioms of vector space does $V$ follow and which it doesn't.
My doubt is lets say if set $V$ is not closed under addition, shall I proceed to check other axioms under addition for set $V$ (commutativity, associativity, existence of identity element, existence of inverse) or other axioms under addition are formed only on the basis of closure property and hence if $V$ is not closed under addition, we can say that it won't follow other axioms under addition??