Not

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`Not`

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!expr

is the logical NOT function. It gives False if expr is True, and True if it is False.

Details

Not[expr] can be input in StandardForm and InputForm as ¬expr. The character can be entered as !, not, or ∖[Not]. »
Not gives symbolic results when necessary, applying various simplification rules to them.
If you are using the Wolfram System with a text‐based front end, then you cannot use the notation !expr for Not[expr] if it appears at the very beginning of a line. In this case, !expr is interpreted as a shell escape. »

Examples

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Basic Examples (3)Summary of the most common use cases

Negate assertions:

In[1]:=1

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https://wolfram.com/xid/0dsma-hby5cu

Out[1]=1

Symbolic negation:

In[1]:=1

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https://wolfram.com/xid/0dsma-cc2cql

Out[1]=1

Enter using not:

In[1]:=1

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https://wolfram.com/xid/0dsma-fthhge

Out[1]=1

Scope (4)Survey of the scope of standard use cases

Double negation simplifies to the identity:

In[1]:=1

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https://wolfram.com/xid/0dsma-fjou75

Out[1]=1

Negate equations and inequalities:

In[1]:=1

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https://wolfram.com/xid/0dsma-gjlsdx

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dsma-ngqjin

Out[2]=2

Negate quantifiers:

In[1]:=1

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https://wolfram.com/xid/0dsma-cp9n0v

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dsma-fsivv6

Out[2]=2

TraditionalForm formatting:

In[1]:=1

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https://wolfram.com/xid/0dsma-8vcqu

Applications (2)Sample problems that can be solved with this function

Negate conditions in Wolfram Language code:

In[1]:=1

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https://wolfram.com/xid/0dsma-c7wxa2

Out[1]=1

Find the area of the complement of a set given by algebraic conditions:

In[1]:=1

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https://wolfram.com/xid/0dsma-fe4ca9

Out[1]=1

This shows the set:

In[2]:=2

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https://wolfram.com/xid/0dsma-c14dt7

Out[2]=2

Properties & Relations (4)Properties of the function, and connections to other functions

Truth table for Not:

In[1]:=1

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https://wolfram.com/xid/0dsma-ib02ro

Out[1]=1

Use BooleanConvert to simplify the negation of Implies:

In[1]:=1

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https://wolfram.com/xid/0dsma-mia1i

Out[1]=1

De Morgan's laws relate And, Or, and Not:

In[1]:=1

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https://wolfram.com/xid/0dsma-d8jn2e

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dsma-okqqa7

Out[2]=2

Negation of the condition for Boole function:

In[1]:=1

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https://wolfram.com/xid/0dsma-jec7qu

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dsma-yhneo

Out[2]=2

Possible Issues (1)Common pitfalls and unexpected behavior

Use parentheses to ensure ! beginning a line is not taken as a shell escape when using a terminal interface:

In[1]:=1

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https://wolfram.com/xid/0dsma-eerbi

Out[1]=1

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