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.

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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.
Transcribed Image Text: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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