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 universe".
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.
For instance, the
picture to the right displays that A is
a subset of B:
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:
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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 ^
D".
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...