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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 happy and Steve is unhappy.” 1.1.7 (4 points) Identify the premises and conclusions of the following arguments and analyze
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