By Herbert S. Bear

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**Extra resources for Lectures on Gleason Parts**

**Example text**

Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output ((¬p ∨ ¬r) ∧ ¬q) ∨ (¬p ∧ (q ∨ r)) from input bits p, q, and r. 3 25 Propositional Equivalences Introduction An important type of step used in a mathematical argument is the replacement of a statement with another statement with the same truth value. Because of this, methods that produce propositions with the same truth value as a given compound proposition are used extensively in the construction of mathematical arguments.

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.

Making telescopes is one of his hobbies. He is also interested in optics and stereo photography. ” Smullyan is the subject of a documentary short film entitled This Film Needs No Title. 2 Applications of Propositional Logic ¬p p q Inverter FIGURE 1 r p∧q p q OR gate AND gate Basic logic gates. p ∧ ¬q p q p∨q p 21 ¬q (p ∧ ¬q) ∨ ¬r ¬r FIGURE 2 A combinatorial circuit. Complicated digital circuits can be constructed from three basic circuits, called gates, shown in Figure 1. The inverter, or NOT gate, takes an input bit p, and produces as output ¬p.