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Unformatted text preview: The partial function f is regarded as undeﬁned on those elements which do not have an image under f . It is sometimes convenient to refer to this undeﬁned value explicitly as ⊥ (pronounced bottom). A partial function from A to B is the same as a function from A to (B + {⊥}). Example The relation R = {(1, a), (3, a)} is a partial function: A
1 2 3 b a B We can see that not every element in A maps to an element in B . Example The binary relation R on R deﬁned by x R y if and only if is a partial function. It is not deﬁned when x is negative. √ x=y Exercise Let A B denote the set of all partial functions from A to B . If A = m and B  = n, what is the cardinality of A B ? 4.3 Properties of Functions Recall that we highlighted certain properties of relations, such as reﬂexivity, symmetry and transitivity. We also highlight certain properties of functions, which will be used to extend our cardinality deﬁnition to inﬁnite sets. D E FI N I T I O N 4 . 5 ( P R O P E RT I E S O F F U N C T I O N S ) Let f : A → B be a function. We deﬁne the following properties on f : 1. f is onto (sometimes called surjective) if and only if every element of B is in the image of f : that is, ∀b ∈ B . ∃a ∈ A. f (a) = b; 2. f is onetoone (sometimes called injective) if and only if for each b ∈ B there is at most one a ∈ A with f (a) = b: that is, ∀a, a ∈ A. f (a) = f (a ) implies a = a ; 3. f is a bijection if and only if f is both onetoone and onto. Notice that the deﬁnition of an onetoone function is equivalent to a 1 = a2 ⇒ f (a1 ) = f (a2 ), and so an onetoone function never repeats values. 33 ...
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 Spring '10
 Koskesh
 Math

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