Def (Function): A binary relation RCX X Y is called a function from X to Y, written f:X →Y if - single-valued: for every x there is at most one y with xfy - total: for every x there is a y such that xfy. X is called the domain and Y is the codomain. Examples: f:R → R, x → x² add: R2 → R, (x, y) → x + y Recall R-1 = {(y,x): xRy} %3D When is f-1 a function? It needs to be single-valued: - for every y there is at most one x s.th. yf-1x, which is the same as xfy. It needs to be total: - for every y there is an x such that yf-lx, which is the same as xfy.
Def (Function): A binary relation RCX X Y is called a function from X to Y, written f:X →Y if - single-valued: for every x there is at most one y with xfy - total: for every x there is a y such that xfy. X is called the domain and Y is the codomain. Examples: f:R → R, x → x² add: R2 → R, (x, y) → x + y Recall R-1 = {(y,x): xRy} %3D When is f-1 a function? It needs to be single-valued: - for every y there is at most one x s.th. yf-1x, which is the same as xfy. It needs to be total: - for every y there is an x such that yf-lx, which is the same as xfy.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help to understand thoroughly the concepts in the image
Transcribed Image Text:More Set Operations
Def: 24 = {X:X C A} is the powerset of A, that is, the set
of all subsets of A. (More traditional notation: P(A)).
Example:
2{1,2,3} = {Ø, {1}, {2}, {3}, {1,2}, {1,3}, {2,3},{1,2,3}}
Theorem: |24| = 214| for all sets A.
More examples:
2{1} = {Ø, {1}}
2° = {Ø} (this is not the same
as just Ø)
%3D
Partitioning
Partition pairs of natural numbers
N× N = {(n, m):n, m are natural numbers}
By their difference, that is by n– m
Transcribed Image Text:Def (Function): A binary relation
RC X × Y_is called a function from
X to Y, written f:X→Y if
- single-valued: for every x there
is at most one y with xfy
- total: for every x there is a y
such that xfy.
X is called
the domain and Y is
the codomain.
Examples:
f:R → R, x → x²
add: R² → R, (x,y) → x + y
Recall R-1 = {(y,x): xRy}
When is f-1 a function? It needs to
be single-valued:
- for every y there is at most one
x s.th. yf-1x, which is the same
as xfy.
It needs to be total:
- for every y there is an x such
that yf-'x, which is the same as
xfy.
We give these properties their own
names:
Definition: We say a function f is
injective (one-to-one) if for every
y there is at most one x s.th. xfy.
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