Midterm I - 3. Fix real numbers a , b , c , and d . Prove...

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MATH 74 MIDTERM 1 Short response [20 points] 1. Answer true or false. [2 points each] (a) If * is a binary operation on a set S , and * is not commutative, then for any a and b in S , it must be that a * b 6 = b * a. (b) The formula a * b = a - b + sin ± πab 2 ² , a, b Z , defines a binary operation * on Z (the set of integers). (c) The formula a b * c d = 1 bd , a b , c d Q , defines a binary operation * on Q (the set of rational numbers). (d) If * is a commutative binary operation on a set S , then for any a , b , and c in S , it must be that ( a * b ) * c = ( c * b ) * a. 2. Consider the binary operation * defined on N by a * b = a b , a, b N . (You can take for granted that this does indeed define a binary operation on N .) Each of the following questions is worth four points. 2(a). Is * associative? Answer yes or no. 2(b). Describe the set of left identity elements for * . 2(c). Describe the set of right identity elements for * . Proofs (20 points each) Instructions for Exercise 3. Assume nothing about addition except the fact that it is a binary operation on R and that it satisfies axioms A1-A3 listed on the handout.
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Unformatted text preview: 3. Fix real numbers a , b , c , and d . Prove that ( a + b ) + ( c + d ) = (( d + c ) + a ) + b. 2 MATH 74 MIDTERM 1 Instructions for 4-6. For the remaining three proofs you may make free use of the basic notational conventions of arithmetic (order of operations, no need for various parentheses, etc). As a general guide, anything above the numbered list of Facts on the handout, or anything that is one or two steps removed from those, may be assumed and used without explicit reference. If you use a listed fact, however, I would like you to cite it explicitly (in words, or by number). 4. Prove that for any n N , we have n j =1 j 3 = n 2 ( n +1) 2 4 . 5. Suppose that x is a rational number and that y is an irrational number. Prove that x + y is irrational. 6. Prove that for any n N , the number 5 2 n-1 + 1 is an integer multiple of 6....
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This note was uploaded on 02/10/2010 for the course MATH 74 taught by Professor Courtney during the Fall '07 term at University of California, Berkeley.

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Midterm I - 3. Fix real numbers a , b , c , and d . Prove...

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