33b - a b c 1 2 3 A B E XAMPLE 4.9 The function f on...

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E XAMPLE 4.6 Let A = { 1 , 2 , 3 } and B = { a,b } . The function f = { (1 ,a ) , (2 ,b ) , (3 ,a ) } is onto, but not one-to-one, as is immediate from its diagram: A B 1 2 3 a b c It is not possible to deFne a one-to-one function from A to B , as there are too many elements in A for them to map uniquely to B . E XAMPLE 4.7 Let A = { a,b } and B = { 1 , 2 , 3 } . The function f = { ( a, 3) , ( b, 1) } is one-to- one, but not onto: a b 1 2 3 It is not possible to deFne an onto function from A to B in this case, as there are not enough elements in A to map to all the elements of B . E XAMPLE 4.8 Let A = { a,b,c } and B = { 1 , 2 , 3 } . The function f = { ( a, 1) , ( b, 3) ,c, 2) } is bijective:
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Unformatted text preview: a b c 1 2 3 A B E XAMPLE 4.9 The function f on natural numbers deFned by f ( x,y ) = x + y is onto but not one-to-one. To prove that f is onto, take an arbitrary n ∈ N . We must Fnd ( m 1 ,m 2 ) ∈ N × N such that f ( m 1 ,m 2 ) = n . This is easy since f ( n, 0) = n +0 = n . To show that f is not one-to-one, we need to produce a counter-example. In other words, we must Fnd ( m 1 ,m 2 ) , ( n 1 ,n 2 ) such that 34...
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