MHB Testing the Properties of a Relation on Nonnegative Real Triples

[flashback]

Let us define a relation a on the set of nonnegative real triples as follows:

(a1, a2, a3) α (b1, b2, b3) if two out of the three following inequalities a1 > b1, a2 > b2, a3 > b3 are satisfied.

a) (3) Test a for Transitivity and Antisymmetry

We call a triple (x, y, z) special if x, y, z are nonnegative and x + y + z = 1.

b) (3) Show that there is a set S of 3 special triples (|S|=3) such that for any given special triple (x, y, z) which is not in S, we can find at least one triple (a, b, c) in S such that (x, y, z) α (a, b, c)

c) (4) Show that there not exists a set S of 3 special triples (|S|=3) such that for any given special triple (x, y, z) (including that in S), we can find at least one triple (a, b, c) in S such that (x, y, z) α (a, b, c)

If anyone could give me some help with this :)
Ty in advance

 

[Evgeny.Makarov]

Let us define a relation a on the set of nonnegative real triples as follows:

(a1, a2, a3) α (b1, b2, b3) if two out of the three following inequalities a1 > b1, a2 > b2, a3 > b3 are satisfied.

I assume it means exactly two inequalities hold and not at least two.

a) (3) Test a for Transitivity and Antisymmetry

For transitivity, consider $(2,3,1)$, $(1,2,3)$ and $(3,1,2)$. For antisymmetry, having both $(x_1,x_2,x_3)\alpha(y_1,y_2,y_3)$ and $(y_1,y_2,y_3)\alpha(x_1,x_2,x_3)$ is impossible since for some pair $i,j$ of indices we must have $x_i>y_i$ and $x_j>y_j$, but for another pair $i',j'$ we must have $y_{i'}>x_{i'}$ and $y_{j'}>x_{j'}$. The sets $\{i,j\}$ and $\{i',j'\}$ have a common element, say, $k$, and then $x_k>y_k$ and $y_k>x_k$.

We call a triple (x, y, z) special if x, y, z are nonnegative and x + y + z = 1.

b) (3) Show that there is a set S of 3 special triples (|S|=3) such that for any given special triple (x, y, z) which is not in S, we can find at least one triple (a, b, c) in S such that (x, y, z) α (a, b, c)

Consider triples containing two zeros.

c) (4) Show that there not exists a set S of 3 special triples (|S|=3) such that for any given special triple (x, y, z) (including that in S), we can find at least one triple (a, b, c) in S such that (x, y, z) α (a, b, c)

You can't have $(0,0,1)\alpha(a,b,c)$ if $(a,b,c)$ is special.

For the future, please read http://mathhelpboards.com/rules/, especially rule #11.

 

[flashback]

Oh i am really sorry i did not know there were rules. Be sure that next time i will be more carefull when writing a post, although this was the original text that my Discrete Mathematics teacher gave me as an assignment. Thanks a lot.
Best regards.