domain 2 codomain In contrast, these relations that are not bijections b b 4 The first function is not bijective because 3 is related to two elements of the domain. The second function is not bijective because 4 is related to zero elements of the domain. The last relation is not even a function because b is associated with zero elements of the domain Bijective functions are also found in the more-familiar world of real-valued functions For example, f(a)=6.c+5 is a bijective function with domain and codomain R For every real number y in the codomain, f(a)=y for exactly one real number a in the domain. On the other hand, f(r)=x2 is not a bijective function. The number 4 in the codomain is related to both 2 and -2 in the domain 1. 3 The bijection Rule If we can pair up all the girls at a dance with all the boys, then there must be an equal number of each. This simple observation generalizes to a powerful counting rule Rule 1(Bijection Rule). If there exists a bijection f: A- B, then A=B In the example, A is the set of boys, B is the set of girls, and the function f defines how they are paired The Bijection Rule acts as a magnifier of counting ability; if you figure out the size of one set, then you can immediately determine the sizes of many other sets via bijections� � � 4 Counting I X f Y a PP 1 PPPP domain �q b PP 2 codomain c P �PPqP PP 3 PPPPq d � 4 e - 5 In contrast, these relations that are not bijections: X Y X Y X Y a - 1 a - 1 a - 1 b c P PPPPPPq PPPPPq d 3 e 3 2 3 4 b c PPPPPPq QQQQQQs d 3 2 3 4 5 b c d e  PPPPPPq 3 3 2 3 4 The first function is not bijective because 3 is related to two elements of the domain. The second function is not bijective because 4 is related to zero elements of the domain. The last relation is not even a function, because b is associated with zero elements of the domain. Bijective functions are also found in the more­familiar world of real­valued functions. For example, f(x) = 6x+5 is a bijective function with domain and codomain R. For every real number y in the codomain, f(x) = y for exactly one real number x in the domain. On the other hand, f(x) = x2 is not a bijective function. The number 4 in the codomain is related to both 2 and ­2 in the domain. 1.3 The Bijection Rule If we can pair up all the girls at a dance with all the boys, then there must be an equal number of each. This simple observation generalizes to a powerful counting rule: Rule 1 (Bijection Rule). If there exists a bijection f : A → B, then |A| = | | B . In the example, A is the set of boys, B is the set of girls, and the function f defines how they are paired. The Bijection Rule acts as a magnifier of counting ability; if you figure out the size of one set, then you can immediately determine the sizes of many other sets via bijections