Graph Theory and Function Problems

[B3NR4Y]

Homework Statement

1. Consider the Cartesian Product A X B, where A, B are finite nonempty sets, each with carnality greater than 1. There are two functions with domain A X B, called projections with mapping rules p₁(a,b) = a and p₂(a,b) = b. What is the target space of p₁? p₂? Are either of p₁, ₂ one to one? Onto?

2. The star graph on n vertices has one vertex adjacent to all other vertices (and no other adjacencies). Conjecture and prove a formula for the number of edges of the star graph on n vertices.

Homework Equations

None that I can think of

The Attempt at a Solution

1. the target space of p₁ is A
and the target space of p₂ is B. Neither are one-to-one because fixing B, there are multiple A's that could go in the first slot, but there would be the same value for p₂. Same logic for p₁ except fixing A.
They're both onto, because the Cartesian product goes through each value in B, or A, and the projection takes each of those and maps it to the target space. Therefore both are onto.

2. I don't know where to begin.

 

[andrewkirk]

Your answer to 1 is correct.

To do 2, start by drawing the star graphs on 2, 3, 4 and 5 vertices. Put the vertex that is connected to all the others at the centre, so that it looks like a star (well, the graphs with 4 and 5 vertices will. Those with 2 and 3 vertices won't). How many edges does each of those three graphs have? What is the pattern?

 

[epenguin]

Unusual. The only thing I found a little problematic and needing some pondering in question 2 was the word 'the'.

 

[B3NR4Y]

What you found difficult or not has no concern to me.

I will go about doing that andrewkirk, thank you!