Given the following two $32-$bit binary unsigned numbers A and $B,$ find the logic expression of the carry flag when A and $B$ are added. The result is $R.$

$A=a_{31}{a}_{30}{a}_{29}cdots{a}_2{a}_1{a}_0$

$B=b_{31}{b}_{30}{b}_{29}cdots{b}_2{b}_1{b}_0$

$R=r_{31}{r}_{30}{r}_{29}cdots{r}_2{r}_1{r}_0$