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Midterm I

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

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

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