Prove that 2 is irrational by using a contrapositive-based proof. Hint: if a number n is rational, then n may be expressed as the ratio of two whole numbers p and q, i.e., n = p/q; furthermore it can be assumed that p and q are in lowest terms. Examples of fractions not in lowest terms: 4/8 and 2/4, whereas 1/2 is in lowest terms.

State a theorem from one of your prior math classes in which: a. the converse is also a theorem. b. the converse is not a theorem. Use the theorems in Table 5.8 to determine if the following are tautologies. a. [(p ^ q) ? ~r] ? q ?~r b. {[(p ?~r) ? ~q] ^ ~q} ? (~p ^ r)

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Show how resolution can be used to solve the King’s Wise Men Problem. Use both the “plain vanilla” branch and bound, and branch and bound with dynamic programming, to get from the start node (S) to the goal node (G) in parts (a) and (b) of Figure 3.31. When all else is equal, explore nodes in alphabetical order.

Use Prolog to solve The King’s Wise Men Problem presented at the beginning of this chapter. Find the mgu in each pair below or state that unification is not possible. a) Wines (x, y) Wines (Chianti, Cabernet). b) Wines (x, x) Wines (Chianti, Cabernet). c) Wines (x, y) Wines (y, x) d) Wines (Best (bottle), Chardonnay) Wines (best (x), y)

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