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# ps02 - MAT 300 Mathematical Structures Problem Set 2...

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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). 1.4.2 Let A = { USA, Germany, China, Australia } , B = { Germany, France, India, Brazil } , and C = { x | x is a country in Europe } . List the elements of the given sets and determine if any are disjoint or subsets of others. (a) A B = { USA, Germany, China, Australia, France, India, Brazil } (b) ( A B ) \ C = { Germany } \ { x | x is in Europe } = (c) ( B C ) \ A = { France } Disjoint: (b) is disjoint from (a) and (c)

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