The tilde ("TILL-duh")
is the wiggly "~" character at the beginning of ~[(B UC) – A]; on your keyboard, the tilde is probably located at
or near the left-hand end of the row of numbers. The tilde, in this context, says that
I now want to find the complement of what I've shaded. There are two kinds of complement
in this problem. The set-subtraction complement in the previous step throws out any overlap
between two given sets. But the kind of complement we see in this step, the "not"
complement, means "throw out everything you have now and take everything else in the
Practically speaking, the "not"
complement with the tilde says to reverse the shading:
While Venn diagrams are commonly used for set intersections,
unions, and complements, they can also be used to show subsets.
As you can see above, a subset
is a set which is entirely contained within another set. For instance, every set in a Venn
diagram is a subset of that diagram's universe.
Venn diagrams can also demonstrate
"disjoint" sets. In the graphic to the right, A and
B are disjoint:
That is, disjoint sets have no overlap; their intersection
is empty. There is a special notation for this "empty set", by the way: "Ø".
(Unless you have an odd computer set-up, the preceding character looks like an "O"
with a forward slash through it. If you're on a PC, you can type this "empty set" character
by holding down the "ALT" key and typing "0216" on the numeric keypad.)
This "Ø" character is pronounced as "the empty set".
An illustration of a use of these set relationships
would be the manner in which some search engines process searches:
If you type "cats AND dogs" into the
search box, a search engine using this syntax (called "Boolean" logic) will return all
web pages that contain both the word "cats" and the word "dogs".
This corresponds to the set "C
If, on the other hand, you type "cats OR
dogs", the search engine will return web pages that contain either the word "cats"
or the word "dog" (or both, because the mathematical meaning of "or" is "inclusive").
This "or" statement corresponds to the set "C U D".
If you type "cats NOT dogs", the search
engine will return pages containing the word "cats", but only after discarding all
the pages which also contain the word "dogs". This corresponds to the set "C – D".
Certain types of word problems are meant to be solved
using Venn diagrams. We'll look at this next...