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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.4: Mathematical Induction
Problem 27E
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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: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
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.
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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