Counting I domain b 2 codomain cde In this example, the domain is the set fa,b, c, d, el and the range is the set Y= (1, 2, 3, 4, 5/. Related elements are joined by an arrow. This relation is a function because every element on the left is related to exactly one element on the right. In graph-theoretic terms, this is a function because every element on the left has degree exactly 1. If r is an element of the domain, then the related element in the codomain is denoted f(). w often say f maps r to f (a). In this case, f maps a to 1, b to 3, c to 4, d to 3 and e to 5 lently, f(a)=l, f(b)=3, f(c)=4, f(d)=3, and f(e)=5 The relations shown below are not functions. 12345 12345 The relation on the left is not a function because a is mapped to two elements of the codomain. The relation on the right is not a function because b and d are mapped to zero elements of the codomain. Tsk-tsk 1.2 Bijections The definition of a function is rather asymmetric; there are restrictions on elements in the domain, but not on elements in the codomain. a bijection or bijective function is a function f: X- Y that maps exactly one element of the domain to each element of the codomain. In graph terms, both domain and codomain elements are required to hav degree exactly 1 in a bijective function. Here is an example of a bijectionCounting I 3 X f Y a - 1 domain b PP 2 codomain PPPPq c  3  P  P1 PPPP d q 4 e - 5 In this example, the domain is the set X = {a, b, c, d, e} and the range is the set Y = {1, 2, 3, 4, 5}. Related elements are joined by an arrow. This relation is a function because every element on the left is related to exactly one element on the right. In graph­theoretic terms, this is a function because every element on the left has degree exactly 1. If x is an element of the domain, then the related element in the codomain is denoted f(x). We often say f maps x to f(x). In this case, f maps a to 1, b to 3, c to 4, d to 3 and e to 5. Equivalently, f(a) = 1, f(b) = 3, f(c) = 4, f(d) = 3, and f(e) = 5. The relations shown below are not functions: X Y X Y a P - PPPPPq b c P P PPPPPq PPPPPq d 1 1 2 3 4 a b c d - PPPPPPq 1 2 3 4 e - 5 e - 5 The relation on the left is not a function because a is mapped to two elements of the codomain. The relation on the right is not a function because b and d are mapped to zero elements of the codomain. Tsk­tsk! 1.2 Bijections The definition of a function is rather asymmetric; there are restrictions on elements in the domain, but not on elements in the codomain. A bijection or bijective function is a function f : X Y → that maps exactly one element of the domain to each element of the codomain. In graph terms, both domain and codomain elements are required to have degree exactly 1 in a bijective function. Here is an example of a bijection: