1. Consider the code below to test H = W,where H and W are binary arrays of length n. for (int i = 0; i < n; i++) if (H[i] != W[i]) return(false); return(true); Complete the following sentences and give a justications for your answer:

(a) #((H, W)-pairs that give false-return value in mth iteration, 1 m n)=.................

(b) #((H, W)-pairs giving false-return) = ................... (c) #((H, W)-pairs giving true-return) = ................... (d) For n = 4, the average #(iterations for all (H, W)-pairs) = ......... (e) For n = 5, the average #(iterations for all (H, W)-pairs) = ......... (f) For n = 6, the average #(iterations for all (H, W)-pairs) = .........

(g) For general n 1, the average #(iterations for all (H, W)-pairs) = .........

2. Give an alternate if-condition for the code in Problem 1 that would also correctly test H = W.

3. Express the set theoretic relationship between H and W,when we think of Hand W representing subsets of an n-set, that is tested by the following code.

for (int i = 0; i < n; i++) if (H[i] != W[i]) return(false); else return(true);

4. Complete the following sentences.

(a) For a given H of size m,0 m n,# (W disjoint from H)=........

(b) #((H, W)-pairs such that HW = and |H|=m)=.......

(c) #((H, W)-pairs such that HW = )=.......

(d) A matching of (H, W)-pairs with HW = and (H, W)-pairs with HW can be obtained as follows .................................................. For example, the pair ({a, b}, {c}) of 1st kind can be matched with the pair ........................ of 2nd kind. (e) How many(H, W)-pairs would give true return-value for the code in Problem 3?

5. What is wrong in the following way of counting (H, W)pairs with HW non-empty? (Here, H and W are subsets of an nset.)

(a) For a given m-subset X,1 m n,#((H, W)-pairs with HW = X)=3^nm.

(b) #((H, W)-pairs with |HW|=m)=C(n, m)3^nm. (c) #((H, W)-pairs with |HW| 1) = n m=1 C(n, m)3^nm =4^n 3^n.