# Linear Algebra/Comparing Set Descriptions

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*This subsection is optional. Later material will not require the work
here.*

## Comparing Set Descriptions

A set can be described in many different ways. Here are two different
descriptions of a single set:

$$\{\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}z\,\big|\, z\in\mathbb{R}\}
\quad\text{and}\quad
\{\begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix}w\,\big|\, w\in\mathbb{R}\}.$$

For instance, this set contains

$$\begin{pmatrix} 5 \\ 10 \\ 15 \end{pmatrix}$$

(take $z=5$ and $w=5/2$) but does not contain

$$\begin{pmatrix} 4 \\ 8 \\ 11 \end{pmatrix}$$

(the first component gives $z=4$ but that clashes with the third
component, similarly the first component gives $w=4/5$ but the third
component gives something different). Here is a third description of the
same set:

$$\{\begin{pmatrix} 3 \\ 6 \\ 9 \end{pmatrix}+\begin{pmatrix} -1 \\ -2 \\ -3 \end{pmatrix}y\,\big|\, y\in\mathbb{R}\}.$$

We need to decide when two descriptions are describing the same set.
More pragmatically stated, how can a person tell when an answer to a
homework question describes the same set as the one described in the
back of the book?

## Set Equality

Sets are equal if and only if they have the same members.
A common way to show that two
sets, $S_1$ and $S_2$, are equal is to show mutual inclusion: any member
of $S_1$ is also in $S_2$, and any member of $S_2$ is also in $S_1$.[^1]

## Exercises


## Footnotes

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[^1]: More information on set equality is in the appendix.
