Different definitions of ring of sets I am studying measure theory now and I have encountered three different definitions of ring of sets (from different sources), here is the list of them:

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*$\mathcal{R}$ is called ring of sets if $\mathcal{A}, \mathcal{B} \in \mathcal{R}$ implies $\mathcal{A}\cup\mathcal{B} \in \mathcal{R}$ and $\mathcal{A}\cap\mathcal{B} \in \mathcal{R};$

*$\mathcal{R}$ is called ring of sets if $\mathcal{A}, \mathcal{B} \in \mathcal{R}$ implies $\mathcal{A}\cup\mathcal{B} \in \mathcal{R}$ and $\mathcal{A}\setminus\mathcal{B} \in \mathcal{R};$

*$\mathcal{R}$ is called ring of sets if $\mathcal{A}, \mathcal{B} \in \mathcal{R}$ implies $\mathcal{A}\cap\mathcal{B} \in \mathcal{R}$ and $\mathcal{A}\;\triangle\;\mathcal{B} \in \mathcal{R};$
It was easy for me to prove that the $2$ and the $3$ are equivalent (if we have $\mathcal{A}\cup\mathcal{B} \in \mathcal{R}$ and $\mathcal{A}\setminus\mathcal{B} \in \mathcal{R}$ than we have $\mathcal{A}\cap\mathcal{B} \in \mathcal{R}$ and $\mathcal{A}\;\triangle\;\mathcal{B} \in \mathcal{R}$ and vice versa), however, it seems like the $1$ isn't to them. From standard identities of set theory it looks like using only $\cap, \cup$ we can't obtain $\triangle, \setminus$. But I have problem in providing more strict proof.
May be there are some examples of collection of sets which satisfy the $1$ definition but not the $2$ (or vice versa)? Or may be there is some general proof?

(Sorry this is a answer, I don't have enough reputation to comment)
To my understanding the first definition is used in order theory and the later in measure theory
