# Solution check : existential quantifier and empty relation

Let $S$ be a set, $\mathcal{R}$ be a relation on $S^{2}$ such that $(\forall x\in S\mid\neg(x\ \mathcal{R}\ x)\ )$, and $\mathbf{I}_S$ the identity relation on $S$.

At the end of a proof, I jump from $$\neg(\exists x,y\in S\mid\langle x,y\rangle\in\mathbf{I}_S\cap\mathcal{R})$$ to $$(\mathbf{I}_S\cap \mathcal{R}=\emptyset).$$ In your opinion, is that too hasty, considering that the teacher did not give a formal definition of the empty set, only saying it is a set empty of any element?

• Given that $\mathbf I_S \cap \mathcal R \subseteq S^2$, I wouldn't say it is a very long leap; actually, it seems difficult to come up with some middle step. – amrsa Feb 20 '18 at 19:19
• I thought the same – Thrasymaque Feb 20 '18 at 19:48