Negation

Proved by: Not Applicable
References: Set Theory Vocabulary
Justifications: Not Applicable

Specializations: Not Applicable
Generalizations: Not Applicable

Negation

1. The opposite of [For all is true]
   is [There exists for which is not true].
   Above, stands for “property.” Symbollically, the sentence is written
   The opposite of is .
   Another standard notation for is , where the bar | means “such that.”
2. The opposite of [There exists for which P(x) is true]
   is [For all is not true].
   Symbolically the same sentence is written
   The opposite of is .

Remark. A mathematical statement is true if and only if its opposite is true.
Another way to say this is that the opposite of a statement being true implies the statement to be true.

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Example: The statement “All eleven-legged alligators are orange with blue spots” is true, since if it were false, then there would exist an eleven-legged alligator that is not orange with blue spots.
The statement “All eleven-legged alligators are black with white stripes” is equally true.