Problem $4.$ Consider a first$-$order logic with the following symbols:

$$ relational : just equality $(=)$; $$ functional: cat$(,)$;

$$ constant: $0,1,$ and e$.$

For this logic, consider the interpretation where

$$ the universe is the set of all finite binary strings;

$=$ maps to the relation which it true iff its two arguments are identical;

$$ cat maps to the function which returns the concatenation xy of its arguments x and y$.0$ maps to the the single$-$symbol string $0$;

$1$ maps to the the single$-$symbol string $1$; and

$$ e maps to the the empty string $\backslash$epsi $.$

We can use this logic to write formulas which represent complex properties of strings. For example, consider the formula

isOdd$($x$)$ :$=($y$)($x$=$cat$($y$,1))$

represents the property that $$the binary number represented by the string x is odd$.$

In each part below, write a formula which represents the given relation. If you wish, you may use a relation that you have already defined in a previous part.

$($a$)$ isEven$($x$),$ for $$the binary number represented by x is even$.($b$)$ isSubstring$($x$,$ y$),$ for $$x is a substring of y$.$

$($c$)$ isPowerOf$2($x$),$ for $$the binary number represented by x is a power of $2.($d$)$ isEqual$($x$,$ y$),$ for $$the binary numbers represented by x and y are equal$.$