0.f. Equiualences 2 l
iequenceof
some onto
)rs leadsto
uotient set.
3quivalence
in defining
tppi.ng. The
.ct elements
(d) Á : N; ø : b means that a ( b.
(") A: N; ø : ó means that b : k¿ for some integer k'
(f) A: the set of all subsetsof {1',2,3}; X = Y means that lxl : lYl'
(g) A : the set of lines in the plane; t = A r;reaÍß æ is perpendicular to 3t'
( h ) Á : l R x I R ;( * , a ) = ( * r , a r ) m e a n st h a t n 2 + a 2 : æ ! + u ? .
-3øt,
( Ð A : l R x I R ; ( r , A ) = ( n t , A r ) m e a n st h a t U - 3 r : 9 1
is an equivalenceon A
that:
show
case,
In
each
x
U.
A:U
and
2. Let U:{1,2,3}
quotient
,4:.
set
and find the
(a) (a,b) : (at,bt) if a * b : at * bt.
( b ) ( o ,b ) : ( a r , ä r ) i f ø b : a t b t .
( c ) ( ø ,b ) : ( ¿ r , ó r ) i f a : a t .
(d) (4, b) : (ør, br) if a - b : at - bt.
3. In each case, show that : is an equivalence on A and find a (well-defined) bijection
o:A=+8.
( a ) , 4 : Z ; m : r ¿m e a n s t h a t m 2 : n 2 ; B : N '
(b),4 : IR.x IR; (ø,a) = (rr,aù means that 12 + a2 : æ?* A?; B : {ø e lR I t > 0}'
(") A: IR.x IR; (r,A) = (*t,Ar) means that U : Atl B : IR..
lR > 0}'
i a ¡ a : l R + x J R . +( *; , g ) = ( s r , a ù m e a ntsh a ta l æ : a t l n t ; B : { n e l z
(e) A : IR;¿ : grmeansthat ø - y e Z; B : {n € R | 0 < t < 1}'
;her we use
a,generates
ell defined
- 7¿is even
by ø([rz]):
,[rz] implies
- r¿is even.
llows. Thus,
re argument
i n are both
at lm]: ln]
a bijection.
rrent way in
n
(f) A : Zi m: r¿ means that, m2 - n2 is even; B : {0, 1}.
4. Find all partitions of A: {L,2,3,4}'
5. Let Pt: {Ct,Cz,"' ,C*} and'Pz: {D1,Dz,"' ,D.) be partitionsof a set '4'
Ø} isalsoapartitionof A'
(a) ShowthatP:{CcltD¡lCnÀD¡t
(b) If =r, :2, âûd : denote the equivalences afforded by Pt, Pz, and P, respectively,
describe = in terms of =1 and :2 .
6. Let : and - be two equivalences on the same set ,4.
(a) If a:or
implies that a-ø1, show that each - equivalence class is partitioned
by the : equivalence classes it contains.
(b) Define ! on Abywriting o,- a7ir.andonly if both@: @1ând aN a\. showthat
ry is an equivalence and describe the ry equivalence classes in terms of the :
and - equivalence classes.
7. In each case, determine whether o : Q+ --+ Q is well defined, where Q+ is the set of
positive rational numbers. Support your ans'wer'
(a)a(ft): n þ) a(Ð :ffi
k) "(h) : m i n (¿)o(#):ffi
8. Define : and - on lR by * : g if æ- a e Z and by a - g if ø g e Q'
(a) Show that = and - are equivalences.
(b) Show that o : IR= + IR- is well defined and onto if c([ø]=) :
mapping a one-to-one?
9. For a mapping a : A --+B, Iet = denote the kernel
equivalence of a and \et g: A + A= denote the
natural mapping. Define
¡e reasons for
o '. A= + B
bY
o(løl) : sçq¡
for all equivalence classes lø] in ,4=.
(a) Show that ø is well defined and one-to-one,
onto if a is onto.
B
A
t
1/
A=
Is the
[r]-.
/