Scalar multiplication is defined generally as a function ,where , - linear space, for which 5 axioms are true:

It is neccessary for any function that defines scalar multiplication, that all of these axioms are true.

I've noticed that some sources offer just 4 axioms - the first and the second is joined into one. Does it mean that the first and the second axiom is equivalent?

If not, then there must be a function for which the first axiom is false, but the remaining ones is true. But I cannot think of such an example.