# s(A) + s(B) = ? Set Theory

$$A:= \{ x^2 | x \in (2,3\sqrt{2}] , x \in \mathbb{R} \textrm{ and } x^2 \in \mathbb{Z}\}$$

$$B:= \{ x^3 | x \in (-2\sqrt[3]{3},3] , x \in \mathbb{R} \textrm{ and } x^3 \in \mathbb{Z}\}$$

s(A) + s(B) = ?

s(A) is number of elements of A set. For example if $\ A = \{a,b,c\}$

s(A) = 3

-
I fixed the TeX for you. – Willie Wong Mar 13 '11 at 14:04
Thank you Willie :) I hope question is clear, now . – Eray Mar 13 '11 at 14:05

Given $x \in (2, 3\sqrt{2}]$, what can $x^2$ be? It must be

$$2^2 = 4 < x^2 \leq 18 = (3\sqrt{2})^2$$

So $A$ comprises of all integers ($x^2 \in \mathbb{Z}$) bigger than 4 and less-than-or-equal-to 18. There are 14 of them.

Given $x \in (-2\sqrt[3]{3}, 3]$, you similarly find that

$$-24 < x^3 \leq 27$$

which tells you that there are $27 - (-24) = 51$ elements in $B$

Add them together you get $s(A) + s(B) = 65$.

-
+1. It is as easy as that. Meanwhile $s(A \cup B) = 51$ and $s(A \cap B) = 14$. I don't really see why someone would want to calculate this sort of thing. – Henry Mar 13 '11 at 14:22
but x must be a $\mathbb{R}$ , so x can be $\ 2\sqrt{3}$ for example. And $\ (2\sqrt{3})^2 = 12$ and $\ 12 \in \mathbb{Z}$ – Eray Mar 13 '11 at 14:40
@Henry: This is surely an exercise in understanding and manipulating basic notation in how things are defined: using $\{... \| ... \}$, implicit definitions $x^2$, and using close and open bounds to ranges. – Mitch Mar 13 '11 at 14:47
@Mitch: Perhaps you are right. I would have been happier if it asked for $s(A)$ and $s(B)$ separately, but it was asking for the sum which made me dubious. – Henry Mar 13 '11 at 15:01
@Eray Alakese: Willie Wong has counted 12 as an element of $A$ (and, incidentally, also of $B$). – Henry Mar 13 '11 at 15:01