( m 1 ,m 2 ) ± = ( n 1 ,n 2 ) , but f ( m 1 ,m 2 ) = f ( n 1 ,n 2 ) . There are many possibili-ties, such as (1 , 0) and (0 , 1) . In fact, since + is commutative, ( m,n ) , ( n,m ) is a counter-example for any m,n . E XAMPLE 4.10 The function f on natural numbers deFned by f ( x ) = x 2 is one-to-one, but the similar function f on integers is not. The function f on integers deFned by f ( x ) = x + 1 is surjective, but the similar function on natural numbers is not. E XAMPLE 4.11 The function f on the real numbers given by f ( x ) = 4 x + 3 is a bijective function. To prove that f is one-to-one, suppose that f ( n 1 ) = f ( n 2 ) , which means that 4 n 1 + 3 = 4 n 2 + 3 . It follows that 4 n 1 = 4 n 2 , and hence n 1 = n 2 . To prove that f is onto, let n be an arbitrary real number. We have f (( n-3) / 4) = n by deFnition of f , and hence f is onto. Since f is both one-to-one and onto, it is bijective. Notice that the function f on the natural numbers given by f ( x ) = x + 3 is one-to-one but
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This note was uploaded on 01/23/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '10 term at École Normale Supérieure.