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Unformatted text preview: 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 < 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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