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International Contests
Czech-Polish-Slovak Match
1996 Czech and Slovak Match
2
2
Part of
1996 Czech and Slovak Match
Problems
(1)
binary operation, (a ⋆ b) ⋆ b= a , a ⋆ (a ⋆ b)= b
Source: Czech and Slovak Match 1996 P2
10/1/2017
Let ⋆ be a binary operation on a nonempty set
M
M
M
. That is, every pair
(
a
,
b
)
∈
M
(a,b) \in M
(
a
,
b
)
∈
M
is assigned an element
a
a
a
⋆
b
b
b
in
M
M
M
. Suppose that ⋆ has the additional property that
(
a
(a
(
a
⋆
b
)
b)
b
)
⋆
b
=
a
b= a
b
=
a
and
a
a
a
⋆
(
a
(a
(
a
⋆
b
)
=
b
b)= b
b
)
=
b
for all
a
,
b
∈
M
a,b \in M
a
,
b
∈
M
. (a) Show that
a
a
a
⋆
b
=
b
b = b
b
=
b
⋆
a
a
a
for all
a
,
b
∈
M
a,b \in M
a
,
b
∈
M
. (b) On which finite sets
M
M
M
does such a binary operation exist?
Binary operation
algebra