# Final - 9 Give an example of a function g Z → Z that is...

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MATH 74 FINAL EXAM [SOME TYPOS CORRECTED] Short response. [6 pts each] 1. Give an example of an associative binary operation * on R that is not commutative. 2. Give an example of a commutative binary operation # on R that is not associative. 3. Let S = { x R : x 6 = 0 } , and deﬁne the binary operation * on S by a * b = ab | a | , a,b S. Describe the set of left identity elements for S . [Reminder: x is a left identity for * if x * y = y for all y S .] 4. With S and * as in problem 3, describe the set of right identity elements for S . [Reminder: x is a right identity for * if y * x = y for all y S .] 5. Let A = { 1 , 2 , 3 , 4 } . Give an example of a function f : A A satisfying the condition f ( f ( f ( x ))) = x, x A, but not the condition f ( f ( x )) = x, x A. Alternatively: assert that there is no such function. 6. Give an example of an integer a satisfying the two conditions for all n Z , if a | n 2 then a | n , D ( a ) has at least four positive elements. Alternatively: assert that there is no such integer. 7. Let S denote the set { c N : the equation 36 x - 300 y = c has inﬁnitely many solutions ( x,y ) Z 2 } . What is the smallest positive element of S ? 8. Compute gcd(239 , 715).
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Unformatted text preview: 9. Give an example of a function g : Z → Z that is injective but not surjective. 10. Let A = { 1 , 2 , 3 , 4 } . Give an example of an element X ∈ P ( A × A ) with the property that for any Y and Z in P ( A ) we have X 6 = Y × Z . Alternatively, assert that there is no such element. [Reminder: P ( B ) is the set of all subsets of B .] Proofs. [10 points each, do exactly 4 of the 5] I. Let a,b,c be positive integers. Suppose that (i) gcd( a,b ) = 1 and (ii) a | bc . Prove that a | c . II. Let n be a positive integer and suppose that 3 | n 2 . Prove that 3 | n . III. Let a,b,k be positive integers. Prove that gcd( ka,kb ) = k gcd( a,b ). IV. Prove that 7 | (2 n +2 + 3 2 n +1 ) for all positive integers n . V. Let a,b,c be positive integers. Prove that gcd(gcd( a,b ) ,c ) ∈ D ( a, gcd( b,c )). [Reminder: D ( n ) = { m ∈ Z : m | n } , and D ( m,n ) = D ( m ) ∩ D ( n ).]...
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