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Homework 8

# Homework 8 - MATH 74 HOMEWORK 8(DUE WEDNESDAY OCTOBER 31...

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MATH 74 HOMEWORK 8 (DUE WEDNESDAY OCTOBER 31) 1(a). #18 on p. 118 in Eccles. 1(b). If f : X Y and g : Y Z are functions, and g f is surjective, is g necessarily surjective? Prove that it is, or give a counterexample. 1(c). If f : X Y and g : Y Z are functions, and g f is surjective, is f necessarily surjective? Prove that it is, or give a counterexample. 2(a). #9.4 on p. 114 of Eccles. 2(b). If f : X Y and g : Y Z are functions, and g f is injective, is f necessarily injective? Prove that it is, or give a counterexample. 2(c). If f : X Y and g : Y Z are functions, and g f is injective, is g necessarily injective? Prove that it is, or give a counterexample. 3. Let f : R 2 R 2 be the function given by f ( x, y ) = ( x + y, x - y ) , ( x, y ) R 2 . 3(a). Is f surjective? Prove that it is, or prove that it is not. (This would involve exhibiting a specific point ( a, b ) R 2 and proving that ( a, b ) 6∈ Im f .) 3(b). Is f injective? Prove that it is, or prove that it is not. 4. Let g : Z 2 Z 2 be the function given by g ( x, y ) = ( x + y, x - y ) , ( x, y ) R 2 . Note that g is given by the same formula as f of the previous problem. You may use any relevant results from the previous problem. 4(a). Is g injective? (Prove that it is, or prove that it is not.)

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