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1. 3 1. Consider the relations Rs = {(m, n)lm ~ n} eN x N = {(m,n)lm ~ n} eN x N. R~ Show that the intersection of Rs and function. R~ is the graph of a function. Identify the 2. (a) Show that the following flow diagram computes Z as x * y, the product of natural numbers x and y. (b) Spell out explicitly all the functions and relations used here. INPUT x~O,y~O true PRINT z false Z:=Z +X y:= y - 1 3. (a) For each of the maps below, indicate whether it is one-to-one, onto, a bijection, or none of these.

4 Context-Free Grammars 2. What languages do the following FSLs accept? 4 Context-Free Grammars We learn to parse sentences of English with trees like the one shown in Figure 10. The structure of these rules may be summarized in the form: S -. N P V P a sentence (S) can take the form of a noun phrase (N P) followed by a verb phrase (VP). N P -. Det Adj N a noun phrase (N P) can take the form of a determiner (Det), possibly followed by an adjective (Adj), followed by a noun (N). VP -. V NP a verb phrase (VP) can take the form of a verb (V) followed by a noun phrase (NP).

L} yields a partial function. 12 Example. Let A = {a, a'}, B = {b}. Then we may tabulate the partial functions from A to B as b b where - indicates that no value is defined. L}. L is a bijection but we do not claim that any particular jj is an isomorphism. 24 1 Sets, Maps, and Relations 13 Corollary. Given finite sets A and B, there are (IBI + l)IAI = (IB u {-L}I)IAI o partial functions from A to B. We see here a powerful role of theory: to enable us to use a change of representation to convert a problem into a simpler one, or into a problem that we already know how to solve.