$2.$

$$

Example $3\mathrm{.}17$ The set X of all binary strings $($strings with only $0$s and $1$s$)$ having the same number of $0$s and $1$s is defined as follows.

B$.$ is in X$.$

R$1.$ If x is in X$,$ so are $1$x$0$ and $0$x$1.$

R$2.$ If x and y are in X$,$ so is xy$.$

Notice that both recursive cases preserve the property of having the same number of $1$s as $0$s$.$ Both of these cases form new strings from old by adding $0$s and $1$s$,$ and they always add the same amount of each.

Strings can be useful in a number of contexts: text in word processing, genetic sequences in bioinformatics, etc. Thinking recursively can help us define operations that manipulate the symbols in a string. For example, the next recursive definition shows how to reverse the order of a string.

See Example $3\mathrm{.}17.$ Give a recursive definition for the set Y of all binary strings with more $0$s than $1$s$.($Hint: Use the set X of Example $3\mathrm{.}17$ in your definition of Y$.)$