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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## This note was uploaded on 01/23/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '10 term at École Normale Supérieure.

Ask a homework question - tutors are online