Use truth tables to determine if each of the following is a tautology, a contradiction, or just satisfiable: a. (p ? q) ? ~p ? ~q b. (~p ^ ~q) ? ~p ? ~q c. (p ? q ? r) = (p ? q) ^ (p ? r) d. p ? (p ? q) e. p = p ? q f. (p _ q) (p _ q) // consult exercises 2 and 3.

The NOR function, denoted by a _ b = ~ (a ? b); i.e., the NOR is true precisely when the (inclusive) OR is false.

a. Give the truth table for the two-input NOR function.

b. Show that the NOR operator can be used to simulate each of the AND, OR, and NOT operators.

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)

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.

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