t's think about the relation between relations and functions! ) Sometimes a relation is a function, with input "the thing on the left" and output "the thing on the ht”. For example, the relation "r" from Section 6.1 Problem 2 (shown below) is a function. What nction is it? 2. The following relations are on {1,3,5}. Let r be the relation æry iff y = x + 2 and s the relation æsy iff x ≤ y. a. List all elements in rs. b. List all elements in sr. c. Illustrate rs and sr via a diagram. d. Is the relation rs equal to the relation sr? ) Sometimes a relation is not a function, for example the relation "s" from Section 6.1 Problem 2. Why it not a function? ) On the other hand, a function will always give you a relation between the domain and codomain. Just e x → f(x) as the relation. Suppose we use the relation → generated by f(x) = x 2 - 1 on domain and domain R. Is this relation symmetric, antisymmetric, transitive, and/or reflexive? ) Is the function f(x) = x 2 - 1 bijective, injective, and/or surjective? ) It turns out that facts about the relation of a function are related to the function! If you don't know hat the function is but you do know the relation x → f(x) is transitive, what do you know about the omain and range? ) If at first you know don't know what the function is but you do know the relation x→ f(x) is reflexive, hat can you then deduce about the function?
t's think about the relation between relations and functions! ) Sometimes a relation is a function, with input "the thing on the left" and output "the thing on the ht”. For example, the relation "r" from Section 6.1 Problem 2 (shown below) is a function. What nction is it? 2. The following relations are on {1,3,5}. Let r be the relation æry iff y = x + 2 and s the relation æsy iff x ≤ y. a. List all elements in rs. b. List all elements in sr. c. Illustrate rs and sr via a diagram. d. Is the relation rs equal to the relation sr? ) Sometimes a relation is not a function, for example the relation "s" from Section 6.1 Problem 2. Why it not a function? ) On the other hand, a function will always give you a relation between the domain and codomain. Just e x → f(x) as the relation. Suppose we use the relation → generated by f(x) = x 2 - 1 on domain and domain R. Is this relation symmetric, antisymmetric, transitive, and/or reflexive? ) Is the function f(x) = x 2 - 1 bijective, injective, and/or surjective? ) It turns out that facts about the relation of a function are related to the function! If you don't know hat the function is but you do know the relation x → f(x) is transitive, what do you know about the omain and range? ) If at first you know don't know what the function is but you do know the relation x→ f(x) is reflexive, hat can you then deduce about the function?
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 me solve this. Please if you can, kindly explain it in details step by step. I am using this to learn for an upcoming finals. Thank you very much.
Transcribed Image Text:Let's think about the relation between relations and functions!
(1) Sometimes a relation is a function, with input "the thing on the left" and output "the thing on the
right". For example, the relation "r" from Section 6.1 Problem 2 (shown below) is a function. What
function is it?
2. The following relations are on {1,3,5}. Let r be the relation xry iff y = x + 2
and s the relation æsy iff x ≤y.
a. List all elements in rs.
b. List all elements in sr.
c. Illustrate rs and sr via a diagram.
d. Is the relation rs equal to the relation sr?
(2) Sometimes a relation is not a function, for example the relation "s" from Section 6.1 Problem 2. Why
is it not a function?
(3) On the other hand, a function will always give you a relation between the domain and codomain. Just
use x → f(x) as the relation. Suppose we use the relation →generated by f(x) = x 2 - 1 on domain and
codomain R. Is this relation symmetric, antisymmetric, transitive, and/or reflexive?
(4) Is the function f(x) = x 2 - 1 bijective, injective, and/or surjective?
(5) It turns out that facts about the relation of a function are related to the function! If you don't know
what the function is but you do know the relation x→ f(x) is transitive, what do you know about the
domain and range?
(6) If at first you know don't know what the function is but you do know the relation x→ f(x) is reflexive,
what can you then deduce about the function?
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Transcribed Image Text:Let's think about the relation between relations and functions!
(1) Sometimes a relation is a function, with input "the thing on the left" and output "the thing on the
right". For example, the relation "r" from Section 6.1 Problem 2 (shown below) is a function. What
function is it?
2. The following relations are on {1,3,5}. Let r be the relation xry iff y = x + 2
and s the relation æsy iff x ≤y.
a. List all elements in rs.
b. List all elements in sr.
c. Illustrate rs and sr via a diagram.
d. Is the relation rs equal to the relation sr?
(2) Sometimes a relation is not a function, for example the relation "s" from Section 6.1 Problem 2. Why
is it not a function?
(3) On the other hand, a function will always give you a relation between the domain and codomain. Just
use x → f(x) as the relation. Suppose we use the relation →generated by f(x) = x 2 - 1 on domain and
codomain R. Is this relation symmetric, antisymmetric, transitive, and/or reflexive?
(4) Is the function f(x) = x 2 - 1 bijective, injective, and/or surjective?
(5) It turns out that facts about the relation of a function are related to the function! If you don't know
what the function is but you do know the relation x→ f(x) is transitive, what do you know about the
domain and range?
(6) If at first you know don't know what the function is but you do know the relation x→ f(x) is reflexive,
what can you then deduce about the function?
Solution
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