MHB Finding the Set of Ordered Triples of a Ternary Relation on A

[trevor]

Given that A = {1, 2, 3 ……..20} and R is a ternary relation on A defined by equation
x² + 4y = z .Find the set of ordered triples of R.

 

[skeeter]

Given that A = {1, 2, 3 ……..20} and R is a ternary relation on A defined by equation
x² + 4y = z .Find the set of ordered triples of R.

spitballin' it ... hope I didn't miss any

(1,1,5), (1,2,9), (1,3,13), (1,4,17)

(2,1,8), (2,2,12), (2,3,16), (2,4,20)

(3,1,13), (3,2,17)

(4,1,20)

 

[Evgeny.Makarov]

Too late... Posting anyway.

We need to find all triples $(x,y,z)$ such that $x^2+4y=z$ and $1\le x,y,z\le 20$. Let's do an exhaustive search.

$x=1,y=1\implies z=5$
$x=1,y=2\implies z=9$
$x=1,y=3\implies z=13$
$x=1,y=4\implies z=17$
$x=1,y=5\implies z=21$; this does not satisfy $z\le 20$.
$x=2,y=1\implies z=8$
$x=2,y=2\implies z=12$
$x=2,y=3\implies z=16$
$x=2,y=4\implies z=20$
$x=3,y=1\implies z=13$
$x=3,y=2\implies z=17$
$x=4,y=1\implies z=20$

Other values of $x$ and $y$ give $z$ that is greater than 20. So the set is
\[
R=\{(1,1,5),(1,2,9),(1,3,13),(1,4,17),(2,1,8),(2,2,12),(2,3,16),(2,4,20),(3,1,13),(3,2,17),(4,1,20)\}.
\]

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