Question 112
Given a positive integer $n \gt 1$. Let $T$ be the set of all ordered triples $(x, y, z)$ where $x, y$ and $z$ are different positive integers and $1 \le x, y, z \le 2n$. A set $A$ containing ordered pairs $(u, v)$ is called 'connected' to $T$ if for all $(x, y, z) \in T$ then
$$
\{(x, y), (x, z), (y, z)\} \cap A \neq \emptyset.
$$
a) Find the number of elements of set $T$.
b) Prove that there exists a set connected to $T$ that has exactly $2n(n-1)$ elements.
c) Prove that every set that connected to $T$ has at least $2n(n-1)$ elements.
$$
\{(x, y), (x, z), (y, z)\} \cap A \neq \emptyset.
$$
a) Find the number of elements of set $T$.
b) Prove that there exists a set connected to $T$ that has exactly $2n(n-1)$ elements.
c) Prove that every set that connected to $T$ has at least $2n(n-1)$ elements.