quiz4_key - COT 3100 Summer 2001 Quiz #4 (Solutions)...

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COT 3100 Summer 2001 Quiz #4 (Solutions) 06/28/2001 1. (4 pts) Let A = { a , b , c }, B = { a , b }, and f = {( a , b ), ( b , b ), ( c , a )}. Then f : A B . What are f ( a ), f ( b ), and f ( c )? f ( a )= b , f ( b )= b , f ( c )= a . 2. (7pts) Let f : R R be defined by the formula: 3 5 2 ) ( + = x x f (R – the set of real numbers). Prove that f is surjective and injective and find a formula for To prove that f is injective we need to show that if f ( x ) = f ( y ), then x = y . So, assume (2 x +5)/3=(2 y +5)/3. It implies that 2 x +5=2 y +5, and x = y ( by trivial algebra). To prove that f is surjective we should demonstrate, that for any y R we can find x R, such that f ( x )= y . So, let y be any element of R and (2 x +5)/3= y . Then by simple algebra we get 2 x +5 = 3 y , x = (3 y - 5)/2. We see, that for any y R we can always find x = (3 y - 5)/2 , x R, such that f ( x ) = [2 (3 y - 5)/2+5]/3=[3 y - 5 + 5]/3 = y . f - 1 ( y
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