document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191392>Question 1191392</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify the following functions using a K-map\r<BR>\n" );
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document.write( "F(X, Y,Z) = X’Y’Z’ +   X’YZ + XY’Z + XYZ\r<BR>\n" );
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document.write( "Can you please explain to me the last homework question step-by-step? Thank you!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823198 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823198\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Standard list of minterms (3 variables)<BR>\n" );
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document.write( "<tr><TD  colspan=3></TD><TD  colspan=2>&nbsp;&nbsp; &nbsp; Minterms</TD></tr><tr><TD >X</TD><TD >Y</TD><TD >Z</TD><TD >Term</TD><TD >Designation</TD></tr><tr><TD >0</TD><TD >0</TD><TD >0</TD><TD >X'Y'Z'</TD><TD >m0</TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >X'Y'Z</TD><TD >m1</TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD >X'YZ'</TD><TD >m2</TD></tr><tr><TD >0</TD><TD >1</TD><TD >1</TD><TD >X'YZ</TD><TD >m3</TD></tr><tr><TD >1</TD><TD >0</TD><TD >0</TD><TD >XY'Z'</TD><TD >m4</TD></tr><tr><TD >1</TD><TD >0</TD><TD >1</TD><TD >XY'Z</TD><TD >m5</TD></tr><tr><TD >1</TD><TD >1</TD><TD >0</TD><TD >XYZ'</TD><TD >m6</TD></tr><tr><TD >1</TD><TD >1</TD><TD >1</TD><TD >XYZ</TD><TD >m7</TD></tr>\n" );
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document.write( "We'll narrow our focus on these specific minterms<BR>\n" );
document.write( "X'Y'Z' = m0<BR>\n" );
document.write( "X'YZ = m3<BR>\n" );
document.write( "XY'Z = m5<BR>\n" );
document.write( "XYZ = m7\r<BR>\n" );
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document.write( "So we could say,<BR>\n" );
document.write( "F(X,Y,Z) = X'Y'Z' + X'YZ + XY'Z + XYZ<BR>\n" );
document.write( "F(X,Y,Z) = m0 + m3 + m5 + m7<BR>\n" );
document.write( "F(X,Y,Z) = Σ(0,3,5,7)\r<BR>\n" );
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document.write( "Let's set up the K-map diagram.<BR>\n" );
document.write( "First we'll need the headers.<BR>\n" );
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document.write( "Note: this diagram may be a bit wordy/verbose, but it might be more descriptive to see how the terms are laid out.\r<BR>\n" );
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document.write( "As figure 1 indicates, we have 0 and 1 along the left side to represent X' and X in that order.<BR>\n" );
document.write( "Then along the top we have the sequence: 00, 01, 11, 10<BR>\n" );
document.write( "It looks like we're counting in binary but not quite. The next term after 01 is not 10 because we want to flip 1 bit only. For more info, check out <a  rel=nofollow HREF= \"https://www.tutorialspoint.com/what-is-gray-code\">gray code sequencing</a>\r<BR>\n" );
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document.write( "Once the headers are in place, we can locate each of the m0 through m7 terms.<BR>\n" );
document.write( "You can use this informal notation to keep track<BR>\n" );
document.write( "m0 = 000<BR>\n" );
document.write( "m1 = 001<BR>\n" );
document.write( "m2 = 010<BR>\n" );
document.write( "m3 = 011<BR>\n" );
document.write( "m4 = 100<BR>\n" );
document.write( "m5 = 101<BR>\n" );
document.write( "m6 = 110<BR>\n" );
document.write( "m7 = 111<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/hYzoKdz.png\"> <BR>\n" );
document.write( "Note: the numbers in the top row (0,1,3,2) and the numbers in the bottom row (4,5,7,6) are separated by a gap of 4.<BR>\n" );
document.write( "Eg: m1 and m5 are in the same column and 1+4 = 5.\r<BR>\n" );
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document.write( "Since you'll probably do a bunch of K-maps with 3 variables, it might be handy to have this somewhere on a reference page for later. \r<BR>\n" );
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document.write( "After the m0 through m7 are filled in the correct boxes, we'll then replace these m values {m0,m3,m5,m7} with 1s. The rest gets replaced with 0.<BR>\n" );
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document.write( "Here's one possible grouping after using these <a  rel=nofollow HREF= \"http://www.ee.surrey.ac.uk/Projects/Labview/minimisation/karrules.html\">K-map grouping rules</a><BR>\n" );
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document.write( "Notice that each group has either 1 item in it or 2 (those values are powers of 2).<BR>\n" );
document.write( "Furthermore, notice that we've grouped things to have the smallest number of groups while also maxing out each possible group size.\r<BR>\n" );
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document.write( "The '1' marked in red represents the term X'Y'Z' aka m0.<BR>\n" );
document.write( "The '1's marked in blue represent m5+m7 = XY'Z+XYZ = XZ(Y'+Y) = XZ*1 = XZ<BR>\n" );
document.write( "The '1's marked in green represent m3+m7 = X'YZ+XYZ = (X'+X)YZ = 1*YZ = YZ<BR>\n" );
document.write( "Group overlapping is possible.\r<BR>\n" );
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document.write( "Therefore, we end up with <font color=red>X'Y'Z'+XZ+YZ which is the final answer</font><BR>\n" );
document.write( "We could rewrite this as X'Y'Z'+Z(X+Y) if you wanted. <BR>\n" );
document.write( "But I'll choose to stick to the sum of products format instead where each factor is a literal.\r<BR>\n" );
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document.write( "Verification table:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://docs.google.com/spreadsheets/d/1woeTbKWYDyTsGeyhfUNQ4GGrDV-XjqehayOzUohHsA4/edit?usp=sharing\">https://docs.google.com/spreadsheets/d/1woeTbKWYDyTsGeyhfUNQ4GGrDV-XjqehayOzUohHsA4/edit?usp=sharing</a><BR>\n" );
document.write( "You don't need a google account to be able to view the spreadsheet.\r<BR>\n" );
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document.write( "A handy calculator to verify the answer:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.charlie-coleman.com/experiments/kmap/\">https://www.charlie-coleman.com/experiments/kmap/</a><BR>\n" );
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