# MandysNotes

Tuesday, 18 March 2014 00:00

## Negation

By  Gideon
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Suppose that the only information you have about a certain animal is that:

"This animal does not have a duck bill."

If A is the set of animals that have a duck bill, and C is the set of all animals, then the set of all animals that do not have a duck bill is the negation of A (with respect to C).

This is the set "not A," written as:

$\lnot A$

Note that if an animal is a member of the set $$\lnot A,$$ then it cannot be a duckling, and it cannot be a platypus.

Next. conside the set of all animals that are not mammals. This is the set:

"not B"

$= \lnot B.$

A member of the set $$\lnot B$$ cannot be a rabbit, and it cannot be a platypus.

The negation of the set "A or B" is the set

"not ( A or B)"

$= \lnot \left ( A \cup B \right).$

It is the set of animals that are not either duck-billed or mammals:

It is illustrated below:

A member of the set $$\lnot \left( A \cap B\right)$$ cannot be a duckling or a rabbit or a platypus.

Finally, the set animals that are not both duck-billed and mammals is the set:

"not ( A and B)"

$= \lnot \left( A \cap B \right).$

It is the set of all animals that are not platypuses.