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What Does the Upside-Down A Mean in Math?

We typically associate Latin script letters (A, B, C, etc.) with the English language. These letters are easy to understand, unlike pesky mathematical Σ’s and π’s. However, letters can have very familiar forms in various disciplines, such as the upside-down A in math (∀).

The ∀ symbol may look like the familiar capital “A” written upside down, but in mathematics (specifically in predicate calculus), the ∀ is a logic symbol or universal quantifier. You can use it in place of “for all.” This means that ∀ is a shorthand character you’ll use when writing proofs, equations, and sets. 

While the ∀ may look intimidating, understanding it is just as easy as understanding ABC. Keep reading, and you’ll learn not only what ∀ is, but also how to use it in your own work.

What Is the Inverted A Called?

The inverted A in Mathematics does not have a standardized name. Many often call it a “turned A.” Bear in mind that this word is associated with both ∀ and ɐ, and it uses the letter both in and out of mathematics. It is the universal quantification symbol — when referring specifically to math/logic.

What Is Universal Quantification?

“Universal quantification” sounds quite intimidating, but things are not always as they appear. If we break down the word, it’ll be easy for you to understand the meaning. 

The word “universal” refers to something being general or applying to all cases.

English derives the word “quantification” from the word “quantity.” The Cambridge definition of quantification is “the act of measuring or judging the size or amount of something” (source).

Therefore, quantifiers are words we use to refer to the quantity or amount of something. You may be familiar with this if you’re a native English speaker, as we understand words describing quantities not specific — not giving a literal amount or precise measurement — as quantifiers.

These words include:

  • All
  • A bit
  • Many 
  • Some 
  • Few
  • None

So, universal quantification is when a quantifier applies to all circumstances. This is why universal quantification symbols stand in place of the words “for all” and “given any.”

If you want to break down other concepts into their parts, I encourage you to invest in the Oxford New Essential Dictionary — available on Amazon. With over 100,000 entries, this dictionary can assist you with any of your learning needs, be it for school, work, or even home use.

There are multiple ways that you can show universal quantification, but the most common way is the turned A.

What Does It Mean When the A Is Upside Down?

As previously established, ∀ is a logic symbol used in proofs, equations, and sets. The symbol ∀ stands in place of the words “for all” and “for any” to prevent constant repetition (source).

f(x)  ∀ a, b, c ∈ NThis means that f(x) holds true when x is equal to a, b, and c (which are natural numbers) because:a, b and c are letters that stand in place of numbers.∈ means “is an element of” (source).N (often written as ) is a symbol for natural numbers (only lowercase letters stand in place of numbers).
r(x)=x∀x r(x)This means that for all values of x, r(x) is true.
c(x)=x+2>xC(-1)= 1+(-1)>-10>-1 (true)C(1)= 1+2>13>1 (true) C(2)= 2+2>14>1 (true)∀x C(x)This means that for all values of x, C(x) is true.

Note: if you wish to learn more about the different categories of numbers, look at the article “What Does the Backwards 3 Symbol Mean?

What Is a Logic Symbol?

Logic symbols are shapes that represent logical concepts. In first-order logic, the most common logic symbols are:

  • Quantifiers: ∀ and ∃.
  • Logical connectives:  ∧, ∨, →, ↔ ,→, etc.
  • Punctuation symbols: (), [], etc.
  • Variables and subscripts: a, b, c,  z0, etc.
  • Equality symbol: =

Understanding the Symbols ∀ and ∃

We have already established that ∀ shows universal quantification. However, that is not the only type of quantification. There is a second type of quantification known as existential quantification. This means that the variable has one (or more) correct values in the formula.

Unlike ∀, ∃ stands in place of “there exists,” “there is at least one,” or “for some.”

For example, if you have the formula x+2<3, you can try to figure out x by swapping it with real numbers (both positive and negative):

f(-1)=1<3 (true)

f(0)=2<3 (true)

f(1)=3<3 (false)

f(2)=4<3 (false)

This means that the statement is only true when x<1. Therefore, only some values of x are correct. You can also write this as:

∃x (f(x) x<1) (there are only some values of x for which f(x) is true).

You may also encounter ∃! in your studies. This is an example of uniqueness quantification where the variable can only have a single value — for example, x+4=5.

What Is First-Order Logic (In Short)?

Predicate calculus falls under a heading known as first-order logic.

First-order logic is not simply a mathematics concept. In fact, you can encounter it in philosophy, computer sciences, and even linguistics. First-order logic consists of predicates. Therefore, to understand first-order logic, we need first to understand predicates.

What Are Predicates?

Predicates are statements — both in English and math — that are neither true nor false. We can only conclude their correctness (or incorrectness) based on the given variables.

For example:

  • X > 5 (this is only true if x is a number larger than 5)
  • X is the capital of the United States (this is only true if X is Washington DC)

You should also understand proposition logic as it is somewhat the basis for predicate logic.

What is Proposition Logic?

Predicate logic is quite different from proposition logic — and the propositions that are a part of it. In proposition logic, every statement is true or false, but never both (source). For example:

  • Stephenie Meyer wrote Twilight. (true)
  • The sun rises from the west. (false)

In math, you can see this in statements such as:

  • 2+2=5 (false)
  • 20>12 (true)

The only major difference between first-order logic and proposition logic is your use of predicates instead of propositions and the lack of relations or quantifiers — note: you’ll want to remember this word.

Image by Jeswin Thomas via Unsplash

Shorthand Notation

Now, in first-order logic, a predicate statement is divided into two parts. This is useful in helping you format your writing in a short, concise way.

Predicate statements consist of:

  1. A subject
  2. A predicate

The subject is the variable, and the predicate is the part of the statement which provides information about the subject.

Let’s try one example — This has an ∃ response, but it will work similarly with a ∀ statement:

  • x>5 (meaning x is greater than 5)

We can show this formula using H(x) — here, H() represents the predicate, and x is the subject/variable.

We use shorthand notations while writing. Without it, you would constantly have to repeat what you’re writing, which both takes time and can look clunky after a while.

So, if you want to determine the values of X, you don’t have to repeat:

  • 1 is greater than 5 ✘ (false)
  • 2 is greater than 5 ✘ (false)
  • 3 is greater than 5 ✘ (false)
  • 4 is greater than 5 ✘ (false)
  • 5 is greater than 5 ✘ (false)
  • 6 is greater than 5 ✓ (true)

Rather, you can say:

∃x (H(x) 5<x)  

This means that there are only some values of x for which H(x) is true. Writing in shorthand is useful, and it can help you get your answer across concisely.

How Do You Type Inverted Symbols?

Below, you’ll find a few strategies for typing inverted symbols, like the upside down A.

Strategy One: Copy-Pasting

This is the easiest strategy for writing down an ∀. All you need is an internet connection.

  1.  Search “universal quantification symbol” on Google. You should see a featured snippet from Wikipedia that says:

“The traditional symbol for the universal quantifier is “∀,” a rotated letter “A,” which stands for “for all” or “all.” The corresponding symbol for the existential quantifier is “∃,” a rotated letter “E,” which stands for “there exists” or “exists” (source).

  1. Highlight the ∀ in the sentence and either press ctrl+c or manually select it by clicking the right mouse button. Your screen should mirror the one below (on a PC):
Image by Sona Digital Media
  1. Go onto the relevant document and right-click to paste the symbol manually. Doing this manually is probably better as it provides more options.

If you do a basic paste (ctrl+v), by default, the ∀ will likely be a different size, format, and font from what you’ve already written previously.

Google Docs

If you are using Google Docs, you can remedy this by selecting “paste without formatting” or ctrl+shift+v, which will merge the letter into the formatting of the previous text.

Image by Sona Digital Media

However, on Microsoft Word, you need to pick either “merge formatting” or “keep text only.” These are the last two boxes you’ll see in the paste section — see the diagram below.

Image by Sona Digital Media

Bear in mind that math symbols are not available in most fonts, so the font may change to a font where the symbol ∀ is available. Changing the font for any following words should be relatively easy using the taskbar.

Using these strategies, you should be able to insert ∀ into any program you desire by copy-pasting.

Strategy Two: Inserting Manually

There are two ways to insert the symbol manually.

If you are in Word, select “insert symbol.” Once you have completed this step, a menu will open. In the menu, change the font to Cambria and the subset to Mathematical Operators. 

Image by Sona Digital Media

You can also use a shortcut by typing 2200 and pressing Alt+X afterward.

If you are using Google Docs, select “Insert Special Characters.” Doing this will take you to a menu similar to the one below.

Image by Sona Digital Media

Match the settings you see above — the tabs should say categories, symbols, and math. There, you will find ∀ (shown circled in red). Click it to add it to your document.

You can also use the “search by keyword (e.g., arrow) or code point” search bar and type in “turned A.” Or, use the drawing function below it.

You should be able to insert it using a similar technique in other programs, so do not despair if you’re writing ∀ in a program that is neither Microsoft Word nor Google Docs.

Strategy Three: Writing Upside-down in General

Sometimes life can get boring, and we are forced to make our own fun. If so, maybe some of you consider writing upside down a break from the monotony of real life. Sure, it has little to do with math, but it can still be enjoyable!

Upside-down writing also has its advantages. You can use it when posting or sending messages on social media or when creating social media passwords.

Doing this is simple as well. Visit the site, and write whatever you want to write in the “Type text, words, letters, or symbols here:” box.

The sentence will then show up reversed and flipped, where you can copy and paste it. You can get rid of the text effects by unclicking the boxes with the effects.

Image by Sona Digital Media

The text looks like this if you copy past it:

˙ɹǝᴉɟᴉʇuɐnb lɐsɹǝʌᴉun ɹo loqɯʎs ɔᴉƃol ɐ sᴉ A ǝɥʇ ‘(snlnɔlɐɔ ǝʇɐɔᴉpǝɹd uᴉ ʎllɐɔᴉɟᴉɔǝds) sɔᴉʇɐɯǝɥʇɐɯ uᴉ ʇnq ‘uʍop ǝpᴉsdn uǝʇʇᴉɹʍ ”∀“ lɐʇᴉdɐɔ ɹɐᴉlᴉɯɐɟ ǝɥʇ ǝʞᴉl ʞool ʎɐɯ loqɯʎs A lɐɔᴉʇɐɯǝɥʇɐɯ ǝɥ┴

Note: Bear in mind that the mathematical symbol ∀ is slightly different from the letter A turned (Ɐ) by itself. If you look at the differences below, you can see that the math symbol ∀ is a little bit shorter.

Image by Sona Digital Media

If you are trying to write an ∀, it’s a good idea to first try and make use of the previous methods rather than relying on this one, both for accuracy and your own convenience.

Final Thoughts

It is undeniable that ∀ plays an instrumental role in mathematics. That is why we need to know when, where, and how to use it.

So, next time you’re working in your math books, remember the powers of ∀ and ∃ in saving you both time and space, whether you’re writing “for all” or “for some.” Even if you forget these symbols, chances are your teachers won’t.