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

## For All

For all

⠈⠯

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

For all x

⠈⠯⠭

### for all, subscripted x

for all, subscripted x

⠈⠯⠰⠭

## There Exists

There 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:

There exists x

⠈⠿⠭

or there exists, subscript x

There exists, subscripted 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

for all, subscripted x, P, open paren, x, closed paren

⠈⠯⠰⠭⠠⠏⠷⠭⠾

### Example 2

there exists, x, P, open paren, x, closed paren

⠈⠿⠭⠠⠏⠷⠭⠾

### Example 3

therefore p

⠠⠡⠏

### Example 4

logical negation of, there exists, x, Q, open paren, x, closed paren

⠈⠱⠈⠿⠭⠠⠟⠷⠭⠾

### Example 5

for all, x, open parenthesis x squared greater than or equal to x closed parenthesis

⠈⠯⠭⠷⠭⠘⠆⠀⠨⠂⠱⠀⠭⠾