Ethen Blackwell

2022-07-18

For a real vector I know we write $a=\left({a}_{1},{a}_{2},{a}_{3}\right)\in {\mathbb{R}}^{3}$, which means ${a}_{1}\in \mathbb{R}$, ${a}_{2}\in \mathbb{R}$ and ${a}_{3}\in \mathbb{R}$
Suppose instead I have a vector $b=\left({b}_{1},{b}_{2},{b}_{3}\right)$ where ${b}_{1}$, ${b}_{2}$ and ${b}_{3}$ only takes the values 0 or 1. What is the correct notation for this?

Shelby Strong

Expert

$\left({b}_{1},{b}_{2},{b}_{3}\right)\in \left\{0,1{\right\}}^{3}$
Description
For a smaller case of only one element, we write $b\in \left\{0,1\right\}$. For a two-element case, we can then write
$\left({b}_{1},{b}_{2}\right)\in \left\{0,1\right\}×\left\{0,1\right\}\equiv \left\{0,1{\right\}}^{2}$
And likewise we get for the three-element case
$\left({b}_{1},{b}_{2},{b}_{3}\right)\in \left\{0,1\right\}×\left\{0,1\right\}×\left\{0,1\right\}\equiv \left\{0,1{\right\}}^{3}$
Hope this helps

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