I have a question regarding the computation on bivariate probability when using copula function.
Let $u=F_X(x)$ and $v=F_T(t)$ be the CDF for marginals of $X$ and $T$ respectively, a joint distribution $F_{XT}$ can be computed using a copula function $C$ which is defined as, $$F_{XT}(x,t) = Pr(X\le x, T\le t) = C(F_X(x),F_T(t)) = C(u,v)$$
As shown in Figure 1 above, the $C(u,v)$ correspond to the probability of Quadrant 1.
Quadrant 3 can be computed as per Nelsen (2006) which is, $$ \bar{C}(u,v) = 1-u-v+C(u,v)$$ or known as the Survival Copula (denoted by $\bar{C}$).
Thus, is it correct for me to say that if we want to find $Pr(X\le x, T>t)$ or $Pr(X>x, T\le t)$, we can simply use the Quadrant 2 and 4 respectively?
i.e, $$Pr(X\le x, T>t) = u-C(u,v)$$
$$Pr(X>x, T\le t) = v-C(u,v)$$
Thank you for the comments.