31 - 2. Let A = {1, 2, 3} and B = {a, b}. Let f ⊆ A × B...

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Unformatted text preview: 2. Let A = {1, 2, 3} and B = {a, b}. Let f ⊆ A × B be defined by f = {(1, a), (1, b), (2, b), (3, a)}. This f is not a well-defined function, since one element of A is related to two elements in B as is evident from the diagram: A B 1 2 3 b a 3. The following are examples of functions with infinite domains and codomains: (b) the function f : N → N defined by f (x) = x 2 ; (a) the function f : N × N → N defined by f (x, y ) = x + y ; (c) the function f : R → R defined by f (x) = x + 3. The binary relation R on the reals defined by x R y if and only if x = y 2 is not a function, since for example 4 relates to both 2 and −2. P R O P O S I T I O N 4 . 3 ( F I N I T E CA R D I N A L I T Y ) Let A → B denote the set of all functions from A to B , where A and B are finite sets. If |A| = m and |B | = n, then |A → B | = n m . Sketch proof For each element of A, there are m independent ways of mapping it to B . You do not need to remember this proof. 4.2 Partial Functions Recall that it is easy to write Haskell functions which return run-time errors on some (or all) arguments. For example, when designing a program P to compute square roots, it is quite reasonable to have P return an error message for negative inputs. We can either regard P as a function which returns the error answer on negative inputs, or we can regard the program as a partial function which is undefined on negative arguments. D E FI N I T I O N 4 . 4 A partial function f from a set A to a set B , written f : A B , is a relation f ⊆ A × B such that just some elements of A are related to unique elements of B ; more formally, it is a relation which satisfies: (a, b1 ) ∈ f ∧ (a, b2 ) ∈ f ⇒ b1 = b2 32 ...
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This note was uploaded on 01/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

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