252 CHAPTER 4 Functions
- Find the first six terms of the sequences defined for n > 0 as:
(a) H(n) = n^2 (n + 1)2/4
(b) G(n) = n -^1
(c) F(n) = (-1)n2n - 3fn - Find the first six terms of the sequences defined as:
(a) H(O) = 0 and H(n) =H(n - 1) + n3 for n > 1
(b) G(O) = 0 and G(n) =2G(n - 1) + I forn > 1
(c) F(0)=2andF(n)=3F(n-1)-n+3forn> 1- Find a recursively defined function that gives the terms of the following sequences:
(a) 2, 5, 8, 11, 14, ...
(b) 3, 6, 12, 24, 48, ... - The formal definition of a sequence was in terms of a function F, with domain either
N or {0, 1 .... n - 1}. (Ifn = 0, then {0, 1 .... n - 1} = 0.) The formal definition of
a subsequence involves a sequence F and a strictly increasing sequence S of elements
of the domain of F. Since S is a sequence, S is, formally, another function as above.
In parts (a) through (e) of Example 7, identify the functions S and F o S as sets of
ordered pairs.- Prove the following:
(a) Theorem 2(a)
(b) Theorem 2(b)
(c) Theorem 2(d)
(d) Theorem 2(e) - Prove the following:
(a) Theorem 3(a)
(b) Theorem 3(c) - Let A and B be nonempty sets, and let F : A - B be a function. Prove that the fol-
lowing are equivalent:
(a) F is onto.
(b) There is a function G : B --* A such that F o G = ldB.
(c) For any set C and for functions H 1 : B -+ C and H 2 : B --- C, if H 1 o F = H 2 o
F, then H 1 = H 2.
17. Let A and B be nonempty sets, and let F : A -+ B be a function. Prove that the fol-
lowing are equivalent:(a) F is1-1.
(b) There is a function G : B --* A such that G o F = IdA.(c) For any set C and for functions H 1 : C -- , A and H2 : C -+ A, if F o H 1 = F o
1H2, then HI = H2.- Let A and B be sets with A 1 , A 2 c A, and let F : A --+ B. Let F(A 1 ) denote {F(x):
x e Ai } for i = 1, 2. Show that:
(a) If A 1 _C A 2 , then F(A1) C F(A 2 ).
(b) F(A1 U A 2 ) = F(A 1 ) U F(A 2 ).