document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=343057>Question 343057</A>: <I><SPAN STYLE=\"word-break: break-all;\"> let A and B be subsets of U with n(A)=44, n(b)=32, n(A')=25 and n(A and B)=24\r<BR>\n" );
document.write( "\n" );
document.write( "find n(AUB')\r<BR>\n" );
document.write( "\n" );
document.write( "I am not sure how to find the compliment of just a number, ie how did they get n(A') </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #245562 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=245562\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I don't know what the compliment of a number is either.  I suppose it is something along the lines of \"Gee, you are a nice round number\" or \"Oh, what handsome prime factors you have\"  or something like that.  All that aside, you aren't asked to find the complement of \"just a number.\"  That is not what <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n(A')\"> means.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A'\"> means the set of everything in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U\"> <i>except</i> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n(A')\"> means the number of elements that are in the set <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A'\">.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 1 in this process is to draw a rather simple Venn diagram.  Draw a rectangle.  Completely inside of the rectangle draw two circles that only partially overlap.  Now you should have mapped out four regions.  A region that is inside the rectangle but not inside of either circle.  A region that is only inside of one of the circles.  A region that is only inside of the other circle.  And finally a region where the two circles overlap.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Label the inside of the rectangle (but outside of the circles) with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U\">, remembering that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U\"> is the set of everything inside of the rectangle.  Label one of the circles <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A\"> and the other <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large B\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now, start in the middle.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\left(A\ \small{\wedge}\Large\ B\right)\"> is the enumeration of the set represented by the piece of the two circles where they overlap.  In otherwords, everything that is in both A and B simultaneously.  We are given that this enumeration is 24, so write a 24 in the little overlap area.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now we were also given that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n(A)\ =\ 44\">.  Since the enumeration of A includes both elements that are only in A as well as elements that are also in B, we can determine that the part of A that is only in A must number 20 elements, because 44 minus 24 equals 20.  Write a 20 into that part of the circle labeled A that doesn't overlap circle B.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Similarly, we can tell that the part of circle B that does not overlap circle A must have an 8 in it.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Next we want to determine the number that goes outside of both circles.  The enumeration of the complement of A is given as 25.  But the complement of A, in otherwords everything in the universe set that is NOT in A includes both this outer area plus the part of circle B containing the 8.  Since we are given that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n(A')\ =\ 25\">, the part of the universe that is neither in A or B must have an enumeration of 25 minus 8 equals 17.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now you are able to add up all of the numbers you see -- 20 plus 24 plus 8 plus 17 to get the total enumeration of this universe set.  And then realize that the union of A and everything that is not B is just the universe set excluding that part of set B that is not shared with set A.  Take the universe total and subtract 8 to get your answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://www.evolvefish.com/fish/media/E-FlyingSpaghettiEmblem.gif\"><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );