ass2 - | A 1 | = | A 2 | and | B 1 | = | B 2 | then | A 1 \...

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ASSIGNMENT 2 - MATH235, FALL 2007 Submit by 16:00, Monday, September 24 (use the designated mailbox in Burnside Hall, 10 th floor). 1. Consider N × N as a rectangular array: (0 , 0) (0 , 1) (0 , 2) (0 , 3) ... (1 , 0) (1 , 1) (1 , 2) (1 , 3) ... (2 , 0) (2 , 1) (2 , 2) (2 , 3) ... (3 , 0) (3 , 1) (3 , 2) (3 , 3) ... . . . Count them using diagonals as follows: 0 1 3 6 10 ... 2 4 7 11 ... 5 8 12 ... 9 13 ... 14 ... . . . This defines a function f : N × N N where f ( m,n ) is the number appearing in the ( m,n ) place. (For example, f (0 , 0) = 0 ,f (3 , 1) = 13 ,f (2 , 2) = 12.) Provide an explicit formula for f (it is what one calls “a polynomial function in the variables m,n ”. It may be a good idea to first find a formula for f (0 ,n )). 2. Prove that if | A 1 | = | A 2 | and | B 1 | = | B 2 | then | A 1 × B 1 | = | A 2 × B 2 | . 3. Prove that | N | = | Q | . (Hint: Show two inequalities; note that there is an easy injection Q Z × Z ). 4. Prove or disprove: if
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Unformatted text preview: | A 1 | = | A 2 | and | B 1 | = | B 2 | then | A 1 \ B 1 | = | A 2 \ B 2 | . 5. Prove that if a product z 1 z 2 of complex numbers is equal to zero then at least one of z 1 ,z 2 is zero. 6. Let f be the complex polynomial f ( x ) = (3 + i ) x 2 + (-2-6 i ) x + 12 . Find a complex number z such that the equation f ( x ) = z has a unique solution (use the formula for solving a quadratic equation). 7. Find the general form of a complex number z such that: (i) z 2 is a real number (i.e., Im( z 2 ) = 0). (ii) z 2 is a purely imaginary number (i.e., Re( z 2 ) = 0). Also, in each of these cases, plot the answer on the complex plane....
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