MATH1061/7861, Mon 14 Sep 2020
Question 1. For each statement, either prove it or find a counterexample.
◦ For all functions f : X → Y and g : Y → Z, if g∘f is onto, then f is onto.
◦ For all functions f : X → Y and g : Y → Z, if g∘f is onto, then g is onto.
◦ For all functions f : X → Y and g : Y → Z, if g∘f is one-to-one, then f is one-to-one. ◦ For all functions f : X → Y and g : Y → Z, if g∘f is one-to-one, then g is one-to-one.
FM
μμ
or codon
ay
Dad
f
DOM
qq.GL t
goof Gc
gaffGI 2
f
not onto
Onto
x
got onto
got f
Counter
C
onto not onto
onto
VyEl Fscex s.tn TED y z
Xgof f onto
z
aoaf sont
gfffest SGD log if so o s g
g
RIKER
fGD e5 not onto
0 if 0 dE
got is onto pl
To pavegontoneedtoshowHzcZ FgcYSHgg3
Suppose
g
gof
tk
is
bye
X
MATT CARDINALITES 061 IN
Don’t define
A Al
13,43 3
Don define 14 141 61 144
41 same IRI
countable uncountable a
174
www 3 tOOpTsOE
f bijection fX
IN 441
sY.mn 1X
NoIs set
0b
X
fred
i
2
bijections
applying
4
t nametags
Yin Eat t.ae s.t gofEd 3
jemima
TGS C Y so
But
121 101 K
7 yet stgg 3
of all sets with same
um
size as I
BAD F INI I I IEwen
M0
NOOO NOOOO MOLO
Sam
O 334
i gfG 3
7 bigger
Hilbert’s
tf Hotel
it
i
i
i
Ei
i
Question 2. Prove that |Z| = |Z ×
fact 17 74
Question 3. True or false? If x1, x2, x3, … is any infinite sequence, then {x1, x2, x3, …} must be: A. Finite
Hot by 1 prime
S
By
B
174
B. Infinite
C. Countable
D. Uncountable
E. Countably infinite
Question 4. Prove that the intervals [0,1] and [6,9] have the same cardinality. Question 5. Prove that the intervals [0,1] and (6,9) have the same cardinality. Question 6. Prove that every infinite subset of N has the same cardinality as N. Question 7. Show that N × N × N is countable.
Challenge. Prove that for any set X, there is no onto function from X to 𝒫(X).
to
I
Midsen
covers
up
here
r