What do different values of $y$ mean when we plot different graphs?

Question

When solving an equation such as $x^2 = 9$ we write $x = 3$ $or$ $x = -3$. Now take the functions $f(x) = 2x$ and $g(x) = 4$ for all $x$. When we plot them on the $xy$ plane we write $y=2x$ and $y=4$. If we were to evaluate them at $x=1$ we have $y=2$ for the first one, and $y=4$ for the second one. Now I would like to ask, is the correct way to write this, $y=2$ $and$ $y=4$, or is it, $y=2$ $or$ $y=4$? I am uncomfortable with saying $y=2$ and $y=4$ as this implies $2=4$ I think.

Answer 1

On the same $xy$-plane, if you plot $y=2x$, and also plot $y=4$, then:

$$\text{the point $(x,y)$ is marked} \iff (y=2x \text{ or } y=4)$$

If you find the intersection of the marked points with "the points where $x=1$", then there are only two points in the intersection:

$$\begin{align*} ((\text{the point $(x,y)$ is marked}) \text{ and } x=1) &\iff ((y=2x\text{ or } y=4)\text{ and } x=1)\\ &\iff ((y=2\text{ or } y=4)\text{ and } x=1)\\ &\implies (y=2\text{ or } y=4) \end{align*}$$

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I find this usage of "and","or" in English analogous to:

• I bought an apple and I bought an orange.
• There's an apple and an orange in my shopping bag.
• If I pick a fruit out of my shopping bag, the fruit is either an apple or an orange.

Answer 2

Using the same variable $y$ in the two equations is ambiguous. The statement $$ y = 2x \text{ and } y = 4 $$ implies $$ x = 2 $$ while the statement $$ (y = 2x \text{ or } y = 4) \text{ and } x = 1 $$ implies $$ y = 2 \text{ or } y = 4 $$

Edit to include this comment from @FilipMilovanović

As for $x^2=9$, you can think of it as $f(x)=x^2$ and $g(x)=9$ - the reason there is an "or" used to describe the solutions is because these two graphs intersect at two places. In general there can be more than 2 solutions, or none.

Answer 3

Let's go back to the basic definition of AND and OR. $P$ AND $Q$ means both $P$ and $Q$ are true. $P$ OR $Q$ means one of them is true. (In the context of computer science, OR means at least one is true, but in the context of mathematics, OR is usually used to imply only one of them is true)

In your example, writing "$y=2$ AND $y=4$" will mean both $y=2$ and $y=4$ are true, which implies $2=4$ as you have said. Writing "$y=2$ OR $y=4$" won't be appropriate either. In this case, you have to specify which functions you are referring to. The correct form will be "$f(x)=2$ AND $g(x)=4$ when $x=1$", because both the statements "$f(x)=2$ when $x=1$" and "$g(x)=4$ when $x=1$" are correct.

Answer 4

Now take the functions $f(x) = 2x$ and $g(x) = 4$ for all $x$. When we plot them on the $xy$-plane we write $y=2x$ and $y=4.$

Clearer to write $\text“y_1=2x$ and $y_2=4\text”,$ that is, $y_1=f(x)$ and $y_2=g(x).$

I would like to ask, is the correct way to write $\quad y=2$ $and$ $y=4,$

or is it $\quad y=2$ $or$ $y=4$?

The former implies that $\text“y\text”$ is non-existent, while the latter implies that $\text“y\text”$ has no more than two possible values. (Notice that in fact $y_1$ has infinitely many possible values, one for each value of $x.$) Neither is quite what you want.

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Reply to the OP's comment

I always see $y$ used twice, is this lazy notation or mathematically wrong? Must we use $y_1$ and $y_2$ ? If we use OR then its clear that $y$ has two possible values for each $x$ . (And the value we chose to look at is dependant on us looking at either $f$ or $g.$)

I don't know that it is wrong, but it is tacit that these two copies of $y$ have separate contexts, representing the vertical coordinates of separate curves. Say, we are investigating in the science lab how three variables S, N and H depend on the independent variable V; in isolation, you could call each of the three dependent variables $y,$ but while discussing them collectively you'd obviously call them $y_1,$ $y_2,$ $y_3$ (else: S, N, H) instead. $\quad$ In short, recycling the label $\text“y\text”$ is conflating separate contexts.

Answer 5

I think it's a bit more nuanced than "OR vs AND".

From a practical software perspective, when you plot two curves you want to distinguish them in some way. So if you plot f(x) red and g(x) green, then the appropriate relation is:

if y == f(x)
then Color(x, y) == Red
else
  if y == g(x)
  then Color(x, y) == Green
  else Color(x, y) == Transparent

where you can read if A then B else C as a shorthand for (NOT A OR B) AND (A OR C).

In contrast, if you use the relation y == f(x) OR y == g(x), then the curves will share the same color (the union of two curves). Correspondingly, AND would translate to the intersection.

Answer 6

An equation (implicitly) describes a collection of points. When you write $y=2x$ you are actually writing \begin{align} \text{all points $(x,y)$ for which $y=2x$ holds true} \end{align} In set builder notation this looks as follows $$\left\{(x,y)\mid y=2x\right\}$$ where you should read the vertical bar as "such that".

When you write two equations you implicitly require two equations to hold true at the same time. You will generally loose points in this process. This process is also called the "intersection" of these two equations. This name might be more obvious if you draw a Venn diagram

The blue circle describes all points for which $A$ holds true. The red circle describes all points for which $B$ holds true. If you write them both down as a system of equations you will get the intersection $A\cap B$, or the middle part in the Venn diagram. You will loose all the points that are outside this intersection.

Let's try to write an example in set builder notation. We have the system of equations $$\cases{y=x^2-8\\y=2x}$$ In set builder notation this becomes

\begin{align} &\left\{(x,y)\mid y=x^2-8\text{ and }y=2x\right\}=\\ &\left\{(x,y)\mid x^2-8=2x\text{ and }y=2x\right\}=\\ &\left\{(x,y)\mid x^2-2x-8=0\text{ and }y=2x\right\}=\\ &\left\{(x,y)\mid (x+2)(x-4)=0\text{ and }y=2x\right\}=\\ &\left\{(x,y)\mid (x=-2\text{ or }x=4)\text{ and }y=2x\right\}=\\ &\left\{(x,y)\mid (x=-2 \text{ and }y=-4)\text{ or }(x=4\text{ and }y=8)\right\}=\\ &\left\{(-2,-4),(4,8)\right\}\\ \end{align} So our final answer is a set containing two points: $(-2,-4)$ and $(4,8)$. Actually, each step in this derivation is the same set of two points. You can check this by plotting every equation that is joined by an "and" an seeing that the two equations intersect in precisely those two points. I should note that wherever you see "or" you are performing a union operation. A union operation creates a larger set. For example the union of $\{1,2,5\}$ and $\{5,6\}$ is $\{1,2,5,6\}$. It "adds" the sets together but without allowing duplicates.

Now to get to your specific question. You were basically asking what happens if you have the following system of equations \begin{align} y&=2x\\ y&=4\\ x&=1 \end{align} If you plot these you will find that there is no point where all these equations are true at the same time. The intersection of these equations is empty. The answer is the empty set $\{\}$ containing no points.