2.26 please!! Definition 2.24.(a) If S is a set and f is a function, then the preimage of S under f is the set{x € domain(f) | f(x) E S}in set-builder notation. We denote the preimage of S under f by f-1(S).(b) If D is a set and f is a function, then the image ofD under f is the set{y € codomain(f) | y = f(x) for some x E D}.We denote the image of D under f by f(D).Exercise 2.25. Suppose f and g are both functions.(a) Determine exactly when the preimage of a set under f is empty.(b) Determine exactly when the image of a set under f is empty.(c) Show, if A and B are both subsets of the codomain of f, then f-'(AUB) = f¯-'(A)Uƒ-1(B).(d) Show, if A and B are both subsets of the codomain of f, then f-'(An B) = f-1(A)n f-1(B).(e) Show, if A and B are both subsets of the domain of f, then f(AUB) = f(A) u ƒ(B).(f) Show, if A and B are both subsets of the domain of f, then f(An B) = f(A) n f(B).(g) Explain why, if C is a subset of the codomain of f and A is a subset of C, then f-1(C\ A) =f-'(C) \ f-1(A).27(h) Explain why, if C is a subset of the domain of f and A is a subset of C, then f(C) \ f(A) is asubset of f(C \ A).(i) Give an example of a function f, a subset of its domain C and a subset A of C for whichf(C\ A) is not a subset of f(C) \ f(A).(i) Explain why f-1(codomain(f)) = domain(f).(k) Explain why f(domain(f)) = range(f).(1) If A is a subset of the domain of f, explain why f-1(f(A)) = A.(m) If A is a subset of the range of f, explain why f(f(A)) = A.(n) If A is a subset of the codomain of f, explain why f(f-1(A)) is a subset of A.(o) Give an example of a function f and subset of its codomain A for which A is not a subset off(f-(A)).(p) Find the domain of fog in terms of the domain of f and g as a preimage.(g) Show, if A is any subset of the codomain of g, then (f o g)-'(A) = g-1(f-'(A)).(r) Show, if A is any subset of the domain of g, then (f o g)(A) = f(g(A)).Exercise 2.26. If f, 9,1, m are functions for which f Ug and lUm are functions, find the domainof (f Ug) o (1 U m) as a union of preimages.

Given information: f, g, l, m, f∪g and l∪mare functions.

Answered: Exercise 2.26. If f, g,l, m are…

Exercise 2.26. If f, g,l, m are functions for which f Ug and lUm are functions, find the domain of (f U g) o (l Um) as a union of preimages.

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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Concept explainers

Correlation

Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.

Linear Correlation

A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.

Regression Analysis

Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.

Question

2.26 please!!

Definition 2.24.
(a) If S is a set and f is a function, then the preimage of S under f is the set
{x € domain(f) | f(x) E S}
in set-builder notation. We denote the preimage of S under f by f-1(S).
(b) If D is a set and f is a function, then the image ofD under f is the set
{y € codomain(f) | y = f(x) for some x E D}.
We denote the image of D under f by f(D).
Exercise 2.25. Suppose f and g are both functions.
(a) Determine exactly when the preimage of a set under f is empty.
(b) Determine exactly when the image of a set under f is empty.
(c) Show, if A and B are both subsets of the codomain of f, then f-'(AUB) = f¯-'(A)Uƒ-1(B).
(d) Show, if A and B are both subsets of the codomain of f, then f-'(An B) = f-1(A)n f-1(B).
(e) Show, if A and B are both subsets of the domain of f, then f(AUB) = f(A) u ƒ(B).
(f) Show, if A and B are both subsets of the domain of f, then f(An B) = f(A) n f(B).
(g) Explain why, if C is a subset of the codomain of f and A is a subset of C, then f-1(C\ A) =
f-'(C) \ f-1(A).
27
(h) Explain why, if C is a subset of the domain of f and A is a subset of C, then f(C) \ f(A) is a
subset of f(C \ A).
(i) Give an example of a function f, a subset of its domain C and a subset A of C for which
f(C\ A) is not a subset of f(C) \ f(A).
(i) Explain why f-1(codomain(f)) = domain(f).
(k) Explain why f(domain(f)) = range(f).
(1) If A is a subset of the domain of f, explain why f-1(f(A)) = A.
(m) If A is a subset of the range of f, explain why f(f(A)) = A.
(n) If A is a subset of the codomain of f, explain why f(f-1(A)) is a subset of A.
(o) Give an example of a function f and subset of its codomain A for which A is not a subset of
f(f-(A)).
(p) Find the domain of fog in terms of the domain of f and g as a preimage.
(g) Show, if A is any subset of the codomain of g, then (f o g)-'(A) = g-1(f-'(A)).
(r) Show, if A is any subset of the domain of g, then (f o g)(A) = f(g(A)).
Exercise 2.26. If f, 9,1, m are functions for which f Ug and lUm are functions, find the domain
of (f Ug) o (1 U m) as a union of preimages.

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