COMPUTER SCIENCEThe pumping lemma says that every regular language has a pumping length p, such that every string in the language can be pumped if it has length p or more. If p is a pumping length for language A, so is any length $p^{\prime} \geq p$. The minimum pumping length for A is the smallest p that is a pumping length for A. For example, if $A=01^{*}$, the minimum pumping length is 2. The reason is that the string s=0 is in A and has length 1 yet s cannot be pumped; but any string in A of length 2 or more contains a 1 and hence can be pumped by dividing it so that x=0, y=1, and z is the rest. For each of the following languages, give the minimum pumping length and justify your answer. $^{A} a$. $0001^{*}$, $^{\mathrm{A}} \mathrm{b}$. $0^{*} 1^{*}$, c. $001 \cup 0^{*} 1^{*}$, $^{\mathrm{A}} \mathrm{d}$ $0^{*} 1^{+} 0^{+} 1^{*} \cup 10^{*} 1$, e. $(01)^{*}$, f. $\varepsilon$, g. $1^{*} 01^{*} 01^{*}$, h. $10\left(11^{*} 0\right)^{*} 0$, i. 1011, j. $\Sigma^{*}$.