For a string of bits $S,$ let $\frac{?}{\text{b}\text{a}\text{r}}(S)$ denote the complementary string obtained by changing all the $1s$ to $0s$ and all the $0s$ to $1s($equivalently$,\frac{?}{\text{b}\text{a}\text{r}}(S)=So+11111$dots $).$ Show that if the DES key $K$ encrypts $P$ to $C,$ then $\frac{?}{\text{b}\text{a}\text{r}}(K)$ encrypts $\frac{?}{\text{b}\text{a}\text{r}}(P)$ to $\frac{?}{\text{b}\text{a}\text{r}}(C).($HINT: This has nothing to do with the structure of the S$-$boxes. To do the problem, just work through the encryption algorithm and show that the input to the S$-$boxes is the same for both encryptions. A key point is that the expansion of $\frac{?}{\text{b}\text{a}\text{r}}(C)$ is the complementary string for the expansion of $C.)$