Write regular expressions that match the following string patterns.

$($a$)(10$ points$)$ Let Sigma $=\{$a$,$b$,$c$,$d$\}$ be the alphabet of the strings. Let L be a languate consist of all strings made up of just as$,$ bs and ds$,$ including the empty string. So abd, aabdab, adabaa etc. should all match, but aababcad shouldn$$t match because it has a c in it$.$ Which regular expression represents L$?$

$($b$)(10$ points$)$ Let $\backslash$Sigma $=\{$a$,$b$,$c$\}$ be the alphabet of the strings. Let L be a language contains all strings that made up of at least two different kinds of symbols where one of them must be a$($so you cannot have a string such as aaa because it has only one kind, a$,$ but you can have a string such as abcabc because it has at least two kinds of it$.$ Which regular expression represents L$?$

$($c$)($bonus $10$ points$)$ Let $\backslash$Sigma $=\{$a$,$b$\}.$ Let L be a language such that all strings made up of as and bs that have an either even number of a or odd number of b$.$ This is OR condition so if it satisfy at least one condition, then it is acceptable.

$($So aabbbis acceptable as there are two as and three bs$,$ and aaabbb is also acceptable as there are odd number of bs even there are odd numbers of as$.$ Also, bb is acceptable as we count $0$ as an even number even there are even number of bs$.$ However, abb is not acceptable since there are odd number

of as and even number of bs$,$ which violoates both conditions.$)$