Does associativity imply closure?

Consider the set $E=\{-1,0,1\}$, under the standard operation of addition on $\bf R$. Obviously it is associative, but is the group closed under addition?

Also, when you state that, “let $+$ be binary operation on set $A$”, you are already assuming that it is closed, since this binary operation is from $A\times A$ to $A$.

Associativity does not imply closure - both are characteristics required of a set to form a group. Both (usually) need to be verified to show that a set forms a group under said binary operation, although it is the binary operation that usually implies the closure property.