http://www.ck12.org Chapter 16. Logic and Set Theory
∼(P∨Q)⇔∼P∧∼Q
The symbol “⇔” works in mathematical logic and set theory the same way “=” works in arithmetic and algebra. In
this case, the negative appears to distribute throughout the or statement by negating each statement individually and
changes the or statement to an and statement. This is calledDe Morgan’s Law.
Example B
Use set theory circles to interpret the negation of a conjunction and explain how the negation of a conjunction can
be written in a different way.
Solution:The shaded portion in the box represents the area that is withinPandwithinQ. In mathematical logic
this is written asP∧Q. In set theory this area is represented similarly asP∩Qwhere the symbol∩stands for
intersection. As before, the notation between mathematical logic and set theory is slightly different. However, the
reasoning about the relationships and the logical implications are identical.
When you negate the statement, you completely switch what is shaded.