# Equation and system in N

• Jun 8th 2009, 12:54 PM
dhiab
Equation and system in N
• Jun 8th 2009, 01:16 PM
VonNemo19
What do you mean by $\displaystyle N^2$?
• Jun 8th 2009, 05:14 PM
Soroban
Hello, dhiab!

These are strange problems . . .

Quote:

Solve in $\displaystyle N^2\!:\quad\begin{array}{ccc}xy & \leq & 2x \\ x+y &=& 4\end{array}$

The line $\displaystyle x + y \;=\:4$ has intercepts (4,0) and (0,4).
Code:

          |         * |           *           | *           |  *           |    *       - - + - - - * - -           |        *           |

We have the inequality: .$\displaystyle xy - 2x \:\leq\:0 \quad\Rightarrow\quad x(y-2) \:\leq\:0$

This is true when $\displaystyle x$ and $\displaystyle (y-2)$ have opposite signs.

There are two cases: .$\displaystyle \begin{Bmatrix}x \:\geq \:0 \\ y \:\leq \:2\end{Bmatrix}\;\text{ and }\;\begin{Bmatrix}x \:\leq\:0 \\ y \:\geq \:2\end{Bmatrix}$

The graph looks like this:
Code:

      ::::::::|       ::::::::|       ::::::::|       ::::::::|     - - - - 2+ - - - - - -               |::::::::::   ------------+:-:-:-:-:----               |::::::::::               |::::::::::               |::::::::::

Together, we have this graph . . .
Code:

      ::::*:::|       ::::::*:|       ::::::::*       ::::::::| *     - - - - 2+ - * - - - -               |:::::*:::::::     ----------+:-:-:-:*:-:-:-               |:::::::::*:::               |:::::::::::*:               |:::::::::::::*

The final graph looks like this:
Code:

          *  |             * |               *               |             2+ - * - - - -               |    *     ----------+-------*------               |        *               |          *               |            *

How can we describe this graph?

Maybe: .$\displaystyle x + y \:=\:4\:\text{ for }(x \leq 0) \cup (x \geq 2)$

• Jun 8th 2009, 09:32 PM
dhiab
Quote:

Originally Posted by VonNemo19
What do you mean by $\displaystyle N^2$?

http://www.mathramz.com/xyz/latexren...efd0b9ff1b.png = N ŚN
• Jun 8th 2009, 10:50 PM
mr fantastic
Quote:

Originally Posted by dhiab

That is, the set of ordered pairs such that each element in the ordered pair is a natural number.
• Jun 9th 2009, 08:46 AM
dhiab
Quote:

Originally Posted by Soroban
Hello, dhiab!

These are strange problems . . .

The line $\displaystyle x + y \;=\:4$ has intercepts (4,0) and (0,4).
Code:

          |         * |           *           | *           |  *           |    *       - - + - - - * - -           |        *           |
We have the inequality: .$\displaystyle xy - 2x \:\leq\:0 \quad\Rightarrow\quad x(y-2) \:\leq\:0$

This is true when $\displaystyle x$ and $\displaystyle (y-2)$ have opposite signs.

There are two cases: .$\displaystyle \begin{Bmatrix}x \:\geq \:0 \\ y \:\leq \:2\end{Bmatrix}\;\text{ and }\;\begin{Bmatrix}x \:\leq\:0 \\ y \:\geq \:2\end{Bmatrix}$

The graph looks like this:
Code:

      ::::::::|       ::::::::|       ::::::::|       ::::::::|     - - - - 2+ - - - - - -               |::::::::::   ------------+:-:-:-:-:----               |::::::::::               |::::::::::               |::::::::::
Together, we have this graph . . .
Code:

      ::::*:::|       ::::::*:|       ::::::::*       ::::::::| *     - - - - 2+ - * - - - -               |:::::*:::::::     ----------+:-:-:-:*:-:-:-               |:::::::::*:::               |:::::::::::*:               |:::::::::::::*
The final graph looks like this:
Code:

          *  |             * |               *               |             2+ - * - - - -               |    *     ----------+-------*------               |        *               |          *               |            *
How can we describe this graph?

Maybe: .$\displaystyle x + y \:=\:4\:\text{ for }(x \leq 0) \cup (x \geq 2)$

Hello : 0n has two cases in question 2 : x=0 and x no zero