16
0. Preliminaries
(a) a : R - lR defined by a(a) :3 - 4n.
(b) a : R + IR.defined by a(a) : L I 12.
(c)a:N--+Ndefinedby
(t+,
if n is odd,
<
if n is even'
|. Ë,
(d) a : Z x Zt --+Q defined by a(n,m) : #.
(e) a : R. -+ lR x IR defined by a(r) : (r i_L,n - L).
(f) a: A x B +.4 defined by a(ø,b): o. (Assume that A+ Ø + B.)
(g) a: A --+Ax B defined by e(ø) : (a,bo), where ó6 e B is fixed and Af
aln):
Ø.
3. Let Ag B I C Aumappings.
(a) If Ba is onto, show that B is onto.
(b) If Ba is one-to-one, show that a is one-to-one.
(c) If Ba is one-to-one and o is onto, show that B is one-to-one.
(d) If Éo is onto and B is one-to-one, show that o is onto.
(e) If fu : B + C satisfies Ba : Blaand o is onto, show tlnat B : Br.
(f) Ifo1 : AB satisfiesBa: Bal and B is one-to-one,show that a:ar.
4. For a : A --+A, show that a2 : 1¿ if and only if a is invertible and a-7 : a.
5. (a) For A-3 A, show that e,2: a if and only if a(æ): ø for ùl CIe a(A).
(b) If A 3 A satisfies d.2 : d, show that a is onto if and only if a is one-to-one.
Describe o in this case.
(c) Let ¿,4 n J! ,4 satisfy .yþ : I,+. If a: þ.t, show that a2 : a.
6. If lÁl à 2 and a: A--+ á satisfiesaB: pa for all B : A- A, prove that a:!e.
7. In each case, verifu that o-1 exists and describe its artion.
(a) o : lR + lR defined by o(ø) : at Iö, where 0 f a €lR and b e lR.
( b ) o : R + { r € R l , > 1 } d e f i n e db y a ( ø ) : ! * n 2 .
(n+L,
if niseven,
(c) e : N + NI defined by o(n) : {
lr-1,ifr¿isodd.
(d) a : Ax B + B x A defined by o(ø,b) : (b,ø).
8. Let A 3
84,4
satisfy Ba:
I¡. If either o is onto or B is one-to-one, show that
each of them is invertible and that each of them is the inverse of the other.
9. Let A3 B 4Á satisfy Ba:|¡.
If ,4 and B are finite sets with l,4l : lBl, show
that aB:LB,
o¿:þ-1 , and B:o-1.
(Compa.reyour answer with the solution of
Example 8.)
10. For A 3 B 4 1, ,ho* that both aB and Ba have inverses if and only if both a and
B have inverses.
11. Let M denote the set of all mappings a:{1,2} --+B. Define 9: M --+ B x B by
p(a) : (a(1), a(2)). Show that g is a bijection and find the action of rp- I .
12. A mapping ô :.4 --+ B is called a constant
map if there exists bs e B such that
ô(o): bs for all ø € A. Show that a mapping õ: AB is constant if and only if
õa:õforallo:A--+A.
13. If l,4l : n and lBl : *, show that there are rn' mappings A + B.
14. Show that the following conditions are equivalent for a mapping o: AB,
where .4 and B are nonempty.
(a) a is one-to-one.
(b) There exists B : B + A such that Ba: !a.
(c) If 7 : C --+ Aand ô : C --+ Asatisfy d.y : aõ,then 7 : [.