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Venn Diagrams & Set Notation (page 3 of 4)

Sections: Introduction, Logic and set notation, Set-notation exercises, Venn word problems


  • Given the following Venn diagram, shade in ~[(B UC) A].
    • three-circle Venn diagram

    As usual when dealing with nested grouping symbols, I'll work from the inside out.  

     

    The union of B and C shades both circles fully:

    circles B and C are fully shaded

      

    Now I'll do the "complement A" part by cutting out the
    overlap with
    A:

      Copyright Elizabeth Stapel 2000-2005 All Rights Reserved
    the overlap of B with A and the overlap of C with A is no longer shaded

      
    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:

      
    the shading is reversed, so B and C are unshaded except for their overlap with A, and everything outside the circles is shaded
        


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:
 

the A circle is contained entirely within the B circle

  
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.
 
Copyright Elizabeth Stapel 2003-2011 All Rights Reserved

Venn diagrams can also demonstrate
"disjoint" sets. In the graphic to the right,
A and B are disjoint:

  
the A and B circles do not overlap

That is, disjoint sets have no overlap; their intersection is empty. There is a special notation for this "empty set", by the way: "". (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 "" 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...

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Cite this article as:

Stapel, Elizabeth. "Venn Diagrams & Set Notation." Purplemath. Available from
    http://www.purplemath.com/modules/venndiag3.htm. Accessed
 

 



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