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mat300ex1sol

mat300ex1sol - MAT 300 Mathematical Structures Exam 1 1(30...

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MAT 300 , Mathematical Structures Name: ANSWERS Exam 1 February 25, 2010 1. (30 points, 5 points each) Short-answer questions. (a) Complete the following definition, making sure to introduce notation and using the appropriate quantifier: For integers a and b , a divides b if there is an integer k such that b = ak . (b) Complete the following definition, making sure to introduce notation and using the appropriate quantifier: For sets A and B , A is a subset of B if for all x A , x B . (c) Negate the following statement: “For every real number x satisfying x 2 - 3 x - 18 = 0, then x is negative or x is greater than 5.” Logical form : x R , ( P Q R ) where P is x 2 - 3 x - 18 = 0, Q is x < 0, and R is x > 5. Logical negation : x R ( ¬ ( P Q R )), or x R ( P ∧ ¬ ( Q R )) (see part (f)) or x R ( P ∧ ¬ Q ∧ ¬ R ) Negation : There exists a real number x satisfying x 2 - 3 x - 18 = 0 and x 6 < 0 and x 6 > 5, or There exists a real number x satisfying x 2 - 3 x - 18 = 0 and 0 x 5. Notes: (1) The negation of “everyone wearing a blue shirt is wearing black pants” is “someone wearing a blue shirt is NOT wearing black pants.” In other words, we are looking for an x that IS a root to the given equation and that does NOT satisfy the given property. (2) Be very careful in negating order relations for real numbers. The negation of x is negative” is “ x is not negative” or “ x is non-negative” or “ x 0”. Similarly, the negation of “ x is greater than 5” is “ x is less than or equal to 5” or “ x 5”.
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