32 - The partial function f is regarded as undefined on...

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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 one-to-one (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 one-to-one and onto. Notice that the definition of an one-to-one function is equivalent to a 1 = a2 ⇒ f (a1 ) = f (a2 ), and so an one-to-one function never repeats values. 33 ...
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