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Unformatted text preview: MAT 300, Mathematical Structures Dr. L. Mantini Problem Set 2 Solutions Spring 2010 Problems 1.3 : 1, 2, 4, 8; 1.4 : 2, 4, 5, 8, 9, 11; 1.5 : 2, 3, 5, 9, 10 1.3.2 (3 points) Analyze the logical forms of the following statements: (a) x and y are men, and either x is taller than y or y is taller than x Solution: Let M ( x ) mean x is a man and let T ( x,y ) mean x is taller than y . Then the logical form is M ( x ) M ( y ) ( T ( x,y ) T ( y,x )). (b) Either x or y has brown eyes, and either x or y has red hair. Solution: Let B ( x ) mean x has brown eyes and let R ( x ) mean x has red hair. Then the logical form is ( B ( x ) B ( y )) ( R ( x ) R ( y )). (c) Either x or y has both brown eyes and red hair. Solution: Using the same sentence labels as in (b), the logical form is ( B ( x ) R ( x )) ( B ( y ) R ( y )) 1.3.4 (3 points) Write definitions using elementhood tests. (a) { 1 , 4 , 9 , 16 , 25 , 36 , 49 ,... } = { n N  n > 0 and n = m 2 for some m N } . (b) { 1 , 2 , 4 , 8 , 16 , 32 , 64 ,... } = { n N  n > 0 and n = 2 m for some m N } (c) { 10 , 11 , 12 ,..., 19 } = { n N  10 n 19 } 1.3.8 What are the truth sets of the following statements? List a few elements if you can. (a) x is a real number and x 2 4 x + 3 = 0 has truth set { 1 , 3 } (b) x is a real number and x 2 2 x + 3 = 0 has the empty set as its truth set, since no real numbers satisfy this equation (c) x is a real number and 5 { y R  x 2 + y 2 < 50 } has truth set { x R  x 2 +25 < 50 } = { x R  x 2 < 25 } = { x R   5 < x < 5 } = ( 5 , 5)....
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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.
 Spring '07
 thieme
 Math, Logic

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