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Undergraduate contests
Miklós Schweitzer
1997 Miklós Schweitzer
6
6
Part of
1997 Miklós Schweitzer
Problems
(1)
infinite family of functions
Source: miklos schweitzer 1997 q6
9/24/2021
Let
κ
\kappa
κ
be an infinite cardinality and let A , B be sets of cardinality
κ
\kappa
κ
. Construct a family
F
\cal F
F
of functions
f
:
A
→
B
f : A \to B
f
:
A
→
B
with cardinality
2
κ
2^\kappa
2
κ
such that for all functions
f
1
,
⋯
,
f
n
∈
F
f_1,\cdots, f_n \in\cal F
f
1
,
⋯
,
f
n
∈
F
and for all
b
1
,
.
.
.
,
b
n
∈
B
b_1 , ..., b_n \in B
b
1
,
...
,
b
n
∈
B
, there exist
a
∈
A
a\in A
a
∈
A
such that
f
1
(
a
)
=
b
1
,
⋯
,
f
n
(
a
)
=
b
n
f_1(a) = b_1,\cdots, f_n(a) = b_n
f
1
(
a
)
=
b
1
,
⋯
,
f
n
(
a
)
=
b
n
.
infinity
function