Is the differential privacy definition with lower and upper bound equivalent to the definition with just an upper bound?

According to Wikipedia, given a randomized algorithm $\mathcal{A}$, two neighboring datasets $D_1, D_2$, a real number $\epsilon > 0$, $\mathcal{A}$ provides $\epsilon$-differential privacy, if
$$
\frac{\mathbb{P}[\mathcal{A}(D_1) \in S]}{\mathbb{P}[\mathcal{A}(D_2) \in S]} \leq e^\epsilon
$$
for all subsets $S$ of $\mathcal{A}$’s support.

However, I've stumbled across a different definition of $\epsilon$-DP, where also a lower bound is required:
$$
e^{-\epsilon} \leq \frac{\mathbb{P}[\mathcal{A}(D_1) \in S]}{\mathbb{P}[\mathcal{A}(D_2) \in S]} \leq e^\epsilon.
$$
Are these two definitions equivalent?

Yes, the definitions are equivalent. That the second implies the first is obvious. To see that the first implies the second, note that the datasets $D_1,D_2$ can be interchanged. So, if

$$
\frac{\mathbb{P}[\mathcal{A}(D_1) \in S]}{\mathbb{P}[\mathcal{A}(D_2) \in S]} \leq e^\epsilon
$$

for all $D_1,D_2$, then in particular

$$
\frac{\mathbb{P}[\mathcal{A}(D'_2) \in S]}{\mathbb{P}[\mathcal{A}(D'_1) \in S]} \leq e^\epsilon,
$$
hence $$
\frac{\mathbb{P}[\mathcal{A}(D'_1) \in S]}{\mathbb{P}[\mathcal{A}(D'_2) \in S]} \geq e^{-\epsilon},
$$
because all values are positive.

StackExchange Computer Science Q#143189, answer score: 5