Summary Modelling Computing Systems Chapter 6 Faron Moller & Georg Struth
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Course
Logic For Computer Science (INFOB1LI)
Institution
Universiteit Utrecht (UU)
Book
Modelling Computing Systems
Logic for Computer Science/Logic for Computer Technology Chapter 6 Summary of the book Modelling Computing Systems written by Faron Moller and Georg Struth. Summary written in English. Using examples and pictures, the substance and theory are clarified. Given at Utrecht University.
A function f from a set A to a set B is an assignment of exactly one element of B to each element of
A. More generally, we write f : A → B to mean f ∈ A → B.
Example:
- Suppose I’m teaching a class with 5 students, S = {Alice, Bob, Carroll, David, Eve }. At the end
of the class, I need to assign marks from 1 to 10 to each student. More precisely, this
determines a function
We write marks(x) = y when a student x is assigned the mark y by the marks function. Crucially, each
student is assigned a single grade. This rules out situations such as: marks(Alice) = 7 and
marks(Alice) = 10. Furthermore, the marks function should assign a mark to every student. That is,
for each student s in S, there is a mark m in {1..10} such that marks(s) = m. A function A → B must
map every element a ∈ A to a single element b ∈ B. In other words f maps each element a of A to an
element b = f(a), which we will also denote by f : a -> b. So there won’t be a output where 2 numbers
are associated with it.
It is possible for a function f : A -> B to assign the same value from B to two different values of A. So
marks(Bob) = 8 and marks(Carroll) = 8.
Given a function f : A → B we introduce the following terminology:
- We call the set A the domain of the function;
- The set B is the codomain of the function;
- If f(a) = b we refer to a as an argument of the function f, and to b as the value of the
function f on argument a.
- If a function takes more than one argument, f : A1 × A2 × A3 … An we refer to the number of
arguments as the arity. Example: f: A x B x C has 3 numbers of arguments
- A function with two arguments is sometimes called a binary function; often we use infix
notation, writing x + y rather than +(x,y).
- The range of f is the subset of B that f can produce: range(f) = {f(a)|a ∈ A}
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