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# Lesson 11.12: Logical Operators For All, There Exists, and Therefore

## For All

$\forall$
⠈⠯

In print, this looks like an up-side-down capital "A."

In braille, this is a dot four followed by dots one, two, three, four, and six.

The "for all" symbol is commonly used in logic and set theory. For example to show, "for all x" you write the "for all" symbol followed by an x or a subscripted x as follows:

### for all x

$\forall x$
⠈⠯⠭

### for all, subscripted x

${\forall }_{x}$
⠈⠯⠰⠭

## There Exists

$\exists$
⠈⠿

In print, this looks like a backwards capital "E."

In braille, this is a dot four, followed by a full cell.

The "there exists" symbol is commonly used in logic and set theory. For example to say, "there exists an element x" as follows:

$\exists x$
⠈⠿⠭

or there exists, subscript x

${\exists }_{x}$
⠈⠿⠰⠭

## Therefore

$\therefore$
⠠⠡

In print, this looks like three triangular dots.

In braille, this is a dot six followed by dots one and six.

### Example 1

${\forall }_{x}P\left(x\right)$
⠈⠯⠰⠭⠠⠏⠷⠭⠾

### Example 2

$\exists \mathrm{xP}\left(x\right)$
⠈⠿⠭⠠⠏⠷⠭⠾

### Example 3

$\therefore p$
⠠⠡⠏

### Example 4

$¬\exists \mathrm{xQ}\left(x\right)$
⠈⠱⠈⠿⠭⠠⠟⠷⠭⠾

### Example 5

$\forall x\left({x}^{2}\ge x\right)$
⠈⠯⠭⠷⠭⠘⠆⠀⠨⠂⠱⠀⠭⠾