CSE390D Notes for Wednesday, 10/16/24
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sets & functions: primarily vocabulary & notation
look at Rosen list of concepts
set is an unordered collection of objects (without duplicates)
many definitions build on the logic notation we've used...for example:
(all x) (x in A -> x in B) == subset (A is subset of B)
(all x) (x in A <-> x in B) == equality
{x | x in A ^ x in B} == intersection
{x | x in A V x in B} == union
{x | x in A ^ x not-in B} == difference
names for sets of interest:
N natural numbers [0, 1, 2, ...]
Z integers [..., -2, -1, 0, 1, 2, ...]
Z+ positive integers [1, 2, 3, ...]
Q rational numbers [p/q] with p in Z, q in Z and q != 0
R real numbers
empty set has no elements (is a subset of all sets)
set complement -- relative to a universal set, U - S
look at set identities
cardinality of a set: number of elements, |S|
power set: set of all subsets of a set
definition:
Let A and B be sets. The Cartesian product of A and B, denoted by
A x B, is the set of all ordered pairs (a, b) where a is in A and
b is in B:
A x B = {(a, b) | a in A ^ b in B}
Functions:
definitions:
Let A and B be nonempty sets. A function f from A to B is an
assignment of exactly one element of B to each element of A. We
write f(a) = b if b is the unique element of B assigned by the
function f to the element a of A. If f is a function from A to B,
we write f : A -> B. Functions are sometimes called mappings.
If f is a function from A to B, we say that A is the domain of f
and B is the codomain of f. If f(a) = b, we say that b is the
image of a and a is the preimage of b. The range, or image, of f
is the set of all images of elements of A. Also, if f is a
function from A to B, we say that f maps A to B.
A function f is said to be one-to-one, or an injection, if and
only if f(a) = f(b) implies that a = b for all a and b in the
domain of f. A function is said to be injective if it is
one-to-one.
A function f from A to B is called onto, or a surjection, if and
only if for every element b in B there is an element a in A with
f(a) = b. A function f is called surjective if it is onto.
using sets {a, b, c, d} and {1, 2, 3, 4}, looked at examples of:
one-to-one, but not onto
onto, but not one-to-one
one-to-one and onto
neither one-to-one nor onto
not a function
exercise:
Function Mapping onto 1-1
f(n)=n+1 Z -> Z yes yes
f(n)=n+1 N -> N no yes
f(n)=n^2 Z -> Z no no
f(n)=n^2 Z -> N no no
definition:
The function f is a one-to-one correspondence, or a bijection, if
it is both one-to-one and onto. We also say that such a function
is bijective.
Let f be a one-to-one correspondence from the set A to the set B.
The inverse function of f is the function that assigns to an
element b belonging to B the unique element of a in A such that
f(a) = b. The inverse function of f is deonted by f-1. Hence,
f-1(b) = a when f(a) = b.
Let g be a function from the set A to the set B and let f be a
function from the set B to the set C. The composition of the
functions f and g, denoted for all a in A by "f o g" is the
function from A to C defined by:
(f o g)(a) = f(g(a)).
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Stuart Reges
Last modified: Fri Oct 18 10:45:51 PDT 2024