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Putnam
1984 Putnam
B3
B3
Part of
1984 Putnam
Problems
(1)
existence of operator on set which is nonassociative and invertible
Source: Putnam 1984 B3
9/2/2021
Prove or disprove the following statement: If
F
F
F
is a finite set with two or more elements, then there exists a binary operation
∗
*
∗
on
F
F
F
such that for all
x
,
y
,
z
x,y,z
x
,
y
,
z
in
F
F
F
,
(
i
)
(\text i)
(
i
)
x
∗
z
=
y
∗
z
x*z=y*z
x
∗
z
=
y
∗
z
implies
x
=
y
x=y
x
=
y
(
ii
)
(\text{ii})
(
ii
)
x
∗
(
y
∗
z
)
≠
(
x
∗
y
)
∗
z
x*(y*z)\ne(x*y)*z
x
∗
(
y
∗
z
)
=
(
x
∗
y
)
∗
z
Sets
abstract algebra