104-hw1 - (i) a 1 = a (ii) a x + y = a x a y (iii) a x y =...

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Math 104 - Lecture 1 - Problem Set 1 Due June 24, 2010 Do the following exercises from the book: 1.8, 1.9. 1.10 1. Recall that in class we defined the equivalence relation on Z × ( Z \ { 0 } ) by ( a, b ) ( c, d ) if ad = bc . We then defined the rational numbers, Q , as the equivalence classes of . Based on this, give a definition of multiplication for Q . Prove that your definition is well-defined. 2. Equivalence Relations . Listed below are some relations on R . Prove or disprove that each is an equivalence relation. If it is an equivalence relation prove that the number of equivalence classes is finite or infinite. (a) a 1 b if | a - b |≤ 7 (b) a 2 b if a - b Q (c) a 3 b if a, b Z 3. Exponentiation . Exponentiation is a common function in mathematics. Properties of exponentiation include:
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Unformatted text preview: (i) a 1 = a (ii) a x + y = a x a y (iii) a x y = ( a x ) y (iv) if a > 1 and x > y then a x > a y . (a) Dene a b when a is a positive real number and b is a nonegative integer. Prove that your denition has the properties above. (b) Use part (a) to dene a b when a is a positive real number and b is an integer. Prove that your denition has the properties above. (c) Use part (b) to dene a b when a is a positive real number and b is a rational number. Prove that your denition has the properties above. (d) Use part (c) to dene a b when a is a positive real number and b is a real number. Prove that your denition has the properties above....
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