Like if we say that in this set identity element is $0$. So, inverse of any element of $G$ is a number that if we add in the that element, it gives result zero : $$1+(x)= 0$$ where $x$ is the inverse of $1$.
What I think is $x$ should be $-1$, which is not in the set. So, it doesn't fullfil the $4$th condition of a group. So, it is not a group but in my exam question, the right answer was group. How is that? Where am I wrong?
A group is not a set. A group is a set together with a binary operation on that set, that satisfies certain axioms. So asking whether $G=\{0,1\}$ is a group is meaningless, unless we also define what the sum of two elements should be.
To answer your question about the exam question we need to know exactly what the question was. Word for word.