Problem $6.[15$ points$]$ For each language, either prove the language is regular $($by creating an implementation with your choice of a DFA, NFA, RegEx, Regular Grammar, or using closure properties$),$ or prove that it is not regular $($by either using the pumping lemma or closure properties$).$

$($b$)\{win\{0,1{\}}^{**}|w=w^R\}$

$($c$)\{uxvx|u,v,$xin$\{a,b{\}}^{+}\}$where $\{a,b{\}}^{+}=\{a,b\}\{a,b{\}}^{**}$

$($e$)\{a^i{b}^j|i,j0,i+j=5\}$

$($f$)\{a^i{b}^j|i,j0,i-j=5\}$

$($g$)\{a^i{b}^j|i,j0|i-j|,-={?}_{5}0\}$