Use of either/or in maths [closed]

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I have been using these two words for a long time, especially when representing the solutions to quadratic equations. But I am little confused. These terms are often used simultaneously, but it seems that I am confused with the three meanings they imply when they are used simultaneously.

Let me add a few examples to add some clarity:

First Meaning:

For example when we say, "Boys will go to the Church either tomorrow or the day after tomorrow and blah blah blah"

Consider another example: that when someone asks how many oranges are in the basket and we reply, "Either 2 or 3, but they are not sweet".

So the first meaning, they simultaneously provide, is that they show choice between two or more things i.e. if the boys will go to the Church then they would go there only tomorrow or if they don't go there tomorrow then they would go there the day after tomorrow i.e., only one option is applicable.

When we say,

$Either$ $x=2$ $or$ $x=3$

Do we mean that Either (only) $x=2$ or (only) $x=3$?

Second Meaning:

I have no example for this meaning other than of "Solutions to Quadratics"

Suppose we say

Either $x=2$ or $x=-2$

Then, do we mean that at a time Either $x=2$ or $x=-2$? However both are solutions to the given equation.

Third Meaning:

For instance when we say, "Randy studies German on either Tuesday or Friday."

Do we represent our answer using Either/or to minimize the ambiguity that is, whether both be the solutions or only either of them?

If you haven't get me yet then feel easy to post a general answer however a general answer would be of more value.

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3 Answers

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What you are getting at is the difference between the inclusive disjunction and the exclusive disjunction.

Suppose you have two arguments, $P$ and $Q$. The following is the truth table for the "inclusive or" often denote by $P\lor Q$:

$ \boxed{ \begin{array}{c|c|c} P & Q & P\lor Q \\ \hline T & T & T\\ T & F & T\\ F & T & T\\ F & F & F \end{array}} $

The following is the truth table for the "exclusive or," sometimes denoted by $P\,\dot\lor\,Q$ (other notations include $\mathsf{XOR},\oplus,\underline\vee, \dot\vee, \nleftrightarrow, \not\equiv$, etc.):

$ \boxed{ \begin{array}{c|c|c} P & Q & P\,\dot\lor\,Q \\ \hline T & T & F\\ T & F & T\\ F & T & T\\ F & F & F \end{array}} $

As you can see, $P\lor Q$ is true when either $P$ is true or $Q$ is true (or both), whereas $P\,\dot\lor\,Q$ is true when either $P$ is true or $Q$ is true (but not both).

That's really all there is to it.

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This is a comment rather than an answer.

Suppose you have a quadratic equation, say

$$x^2-x-6=0.$$

To solve this we first assume that $x$ is a solution. From this assumption we derive that $x=-2$ or $x=-3$.

So $x$ a solution $\Rightarrow x=-2$ or $x=3$.

Therefore the set of numbers that solve the equation are $\{-2,3\}$ --- and we can say that $-2$ and $3$ are solutions.

Our solution technique, assuming $x$ is a solution, can only give us one number at a time but if there 'can be more than one time'.

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To answer directly to your question and add to the previous answer, when formalizing a statement such as:

• "the solution for the equation $x^2-1=0$ is either $x=1$ or $x=-1$"

What we are saying formally in Mathematics is (assuming we are working with real numbers):

• "for every real number $x$, if $x$ satisfies that $x^2-1=0$, then $x=1$ or $x=2$"

And completely formalized in first order logic we would get:

• $\forall x[x^2-1=0\Rightarrow (x=1\vee x=-1)]$ (i.e. we are using the inclusive disjunction)

But what does this symbol $\vee$ really means? From the options that you listed the second meaning is the one closest to it and that's the one you should generally assume when dealing with mathematic statements. Whenever we do not intend that meaning, i.e. we are using the exclusive disjunction, it is common to say "either + or + but not both".

Something similar happens when using "if+then" statements: sometimes we use those kind of statements in every day English to express causality (x causes y) but in Mathematics that is not their meaning. That's why you should take the logical connectives as different kinds of "either+or, and, if+then". I'll allow myself to promote my videos if you want to watch some further examples:

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