Chapter 1: Q2SE (page 111)
Find the truth table of the compound proposition\((p \vee q) \to (p \wedge \neg r)\).
The truth table can be given with the true values corresponding to the proposition.
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What is the value of x after each of these statements is encountered in a computer program, if x = 1 before the statement is reached?
a) if then
b) if OR then
c) if AND then
d) if XOR then
e) if x < 2 then
Which of these sentences are propositions? What are the truth values of those that are propositions?
A says “We are both knaves” and B says nothing. Exercises 24–31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by Smullyan [Sm78]) who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of other two is. For each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. When there is no unique solution, list all possible solutions or state that there are no solutions.
A says “I am the knight,” B says “A is telling the truth,” and C says “I am the spy.”
Let andbe the statements “xis a professor,” “xis ignorant,” and “xis vain,” respectively. Express each of these statements using quantifiers; logical connectives; andandwhere the domain consists of all people.
a) No professors are ignorant.
b) All ignorant people are vain.
c) No professors are vain.
d) Does (c) follow from (a) and (b)?
Find a compound proposition involving the propositional variables, and r that is true when exactly two of, and r are true and is false otherwise. [Hint: Form a disjunction of conjunctions. Include a conjunction for each combination of values for which the compound proposition is true. Each conjunction should include each of the three propositional variables or its negations.]
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