Answer to Question #349854 in Discrete Mathematics for mengal

Question

The following sets have been defined using the | notation. Re-write them by listing some of the elements.

i. {p | p is a capital city, p is in Europe}

ii. {z | 3z = n2, z and n are natural numbers}

Answer 1

i. {p | p is a capital city, p is in Europe}={Athens, Berlin, Copenhagen, Dublin, London, Madrid, Paris, Oslo, Riga, Rome, Vienna, ...}

ii. {$z | 3z = n^2,$ z and n are natural numbers}

We have: $z=\frac{n^2}{3}.$

As z and n are natural numbers, n is divizible by 3.

At n=3: $z=\frac{3^2}{3}=3,$

at n=6: $z=\frac{6^2}{3}=12,$

at n=9: $z=\frac{9^2}{3}=27,$

at n=12: $z=\frac{12^2}{3}=48, ...$

So

{$z | 3z = n^2,$ z and n are natural numbers} = {3, 12, 27, 48, ...}