ps01 - MAT 300, Mathematical Structures Problem Set 1...

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MAT 300, Mathematical Structures Dr. L. Mantini Problem Set 1 Solutions Spring 2010 1.1 : 2, 4, 6, 7; 1.2 : 2, 4, 8, 12, 13, 17. 1.1.2 (3 points) Analyze the logical forms of the following statements: (a) Either John and Bill are both telling the truth, or neither of them is. Solution : (J B) ( ¬ J ∧ ¬ B), where J: John is telling the truth. B: Bill is telling the truth. (b) I’ll have either fish or chicken, but I won’t have both fish and mashed potatoes. Solution : (F C) ∧ ¬ (F M), where F: I’ll have fish. C: I’ll have chicken. M: I’ll have mashed potatoes. (c) 3 is a common divisor of 6, 9, and 15. Solution : Here we use an abbreviation: we let TD( n ) denote the sentence “3 divides n .” Then our sentence is TD(6) TD(9) TD(15). 1.1.4 Which of the following expressions are well-formed formulas? (a) ¬ ( ¬ P ∨ ¬ ¬ R): well-formed (b) ¬ (P, Q, R): not well-formed (c) P ∧ ¬ P, well-formed (also false—a contradiction!) (d) (P Q)(P R): not well-formed. 1.1.6 Let S stand for the statement “Steve is happy” and G for “George is happy.” What English sentences are represented by the following expressions? (a) (S G) ( ¬ S ∨ ¬ G): “Either Steve or George is happy, and either Steve or George is unhappy,” or “One of Steve or George is happy, and one is unhappy.” (b) (S [G ∧ ¬ S]) ∨ ¬ G: “Steve is happy, or George is happy and Steve is unhappy, or George is unhappy.” (Note that this sentence is a tautology.) (c) S [G ( ¬ S ∨ ¬ G)]: “Steve is happy, or George is happy and one of Steve or George is unhappy.” This could also be said as “Steve is happy, or George is
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This note was uploaded on 04/04/2010 for the course MAT 300 taught by Professor Thieme during the Spring '07 term at ASU.

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ps01 - MAT 300, Mathematical Structures Problem Set 1...

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