# Set Theory................Super-Problem.

• Oct 23rd 2010, 08:31 AM
Arka
Set Theory................Super-Problem.
S is the set of all (h, k) with h, k non-negative integers such
that h + k < n. Each element of S is colored red or blue, so that if (h, k)
is red and h ≤ h, k ≤ k, then (h , k ) is also red. A type 1 subset of S has
n blue elements with diﬀerent ﬁrst member and a type 2 subset of S has n
blue elements with diﬀerent second member. Show that there are the same
number of type 1 and type 2 subsets.

• Oct 23rd 2010, 09:03 AM
Plato
Quote:

Originally Posted by Arka
S is the set of all (h, k) with h, k non-negative integers such
that h + k < n. Each element of S is colored red or blue, so that if (h, k)
is red and h ≤ h, k ≤ k, then (h , k ) is also red
. A type 1 subset of S has n blue elements with diﬀerent ﬁrst member and a type 2 subset of S has n blue elements with diﬀerent second member. Show that there are the same
number of type 1 and type 2 subsets.

It is always true that $h\le h~\&~k\le k$.