1. ## Mappings problem

Hi,
Can anyone offer advice on this problem below please - the text book I'm working with doesn't have any problems like this and I need a solution asap. Thanks in advance.

Construct a mapping B:N → N (natural numbers) such that the equation
B(n)=3nē has exactly three solutions.
For which elements n E N is the solution solved? (solutions must be in N)

2. Originally Posted by jackiemoon
Hi,
Can anyone offer advice on this problem below please - the text book I'm working with doesn't have any problems like this and I need a solution asap. Thanks in advance.

Construct a mapping B:N → N (natural numbers) such that the equation
B(n)=3nē has exactly three solutions.
For which elements n E N is the solution solved? (solutions must be in N)
Define $B(n) = (n-1)(n-2)(n-3) + 3n^2$

3. Hey man thanks for the reply. Could you explain a little how you got the answer. Sorry to be a pain!

4. Originally Posted by jackiemoon
Hey man thanks for the reply. Could you explain a little how you got the answer. Sorry to be a pain!
There are many answers to this question. I realized that $(n-1)(n-2)(n-3)=0$ has exactly three solutions. Thus, I added $3n^2$ to it so that in the equation $(n-1)(n-2)(n-3)+3n^2 = 3n^2$ it cancels and we are left with $(n-1)(n-2)(n-3)=0$.

5. Thanks again. Kinda makes sense now.

I'm stuck on one final question if anyone can help.

"A set P is said to be smaller than a set Q if there exists a mapping from P to Q, but there does not exist a mapping from Q to P. The notation P < Q is used to denote that P is smaller than Q according to this definition. Prove that, given sets P, Q, and R, if P<Q and Q<R, then P<R."

Any tips on how to solve this?

Muchos gracias amigos.

6. Originally Posted by jackiemoon
"A set P is said to be smaller than a set Q if there exists a mapping from P to Q, but there does not exist a mapping from Q to P. The notation P < Q is used to denote that P is smaller than Q according to this definition. Prove that, given sets P, Q, and R, if P<Q and Q<R, then P<R."
First of all what is in red above is false.
It should read: "A set P is said to be smaller than a set Q if there exists an injective mapping from P to Q, but there does not exist an injective mapping mapping from Q to P.
There is always a mapping from Q to P but cannot be injective if Q is more numerous than P.
The proof of the question just involves composition of mappings.