Linear Algebra/Comparing Set Descriptions

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:

$\{\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)z|z\in ℝ\}\text{and}\{\left(\begin{array}{c}2\\ 4\\ 6\end{array}\right)w|w\in ℝ\}.$

For instance, this set contains

$\left(\begin{array}{c}5\\ 10\\ 15\end{array}\right)$

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

$\left(\begin{array}{c}4\\ 8\\ 11\end{array}\right)$

(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:

$\{\left(\begin{array}{c}3\\ 6\\ 9\end{array}\right)+\left(\begin{array}{c}-1\\ -2\\ -3\end{array}\right)y|y\in ℝ\}.$

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. Template:AnchorA 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 $S_1$.

Exercises

/Solutions/