document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191384>Question 1191384</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) = X + X’Y\r<BR>\n" );
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document.write( "Could you please explain me this 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 #823196 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=823196\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Thank you for posting one question at a time. I appreciate it.\r<BR>\n" );
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document.write( "Here are the standard minterms for 2 variables only.<BR>\n" );
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document.write( "<tr><TD colspan = 2></TD><TD  colspan = 2><center>Minterms</center></TD></tr><tr><TD >X</TD><TD >Y</TD><TD >Term</TD><TD >Designation</TD></tr><tr><TD >0</TD><TD >0</TD><TD >X'Y'</TD><TD >m0</TD></tr><tr><TD >0</TD><TD >1</TD><TD >X'Y</TD><TD >m1</TD></tr><tr><TD >1</TD><TD >0</TD><TD >XY'</TD><TD >m2</TD></tr><tr><TD >1</TD><TD >1</TD><TD >XY</TD><TD >m3</TD></tr>\n" );
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document.write( "The given boolean function<BR>\n" );
document.write( "F(X,Y) = X + X'Y<BR>\n" );
document.write( "needs to be rewritten in such a way that we involve <u>only</u> the minterms listed.\r<BR>\n" );
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document.write( "That lone X needs to have a Y or Y' attached to it somehow.\r<BR>\n" );
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document.write( "We have to do a bit of trickery to get the function to be a sum of standard minterms.<BR>\n" );
document.write( "F(X,Y) = X + X'Y<BR>\n" );
document.write( "F(X,Y) = X*1 + X'Y .... identity law<BR>\n" );
document.write( "F(X,Y) = X*(Y+Y') + X'Y .... inverse law<BR>\n" );
document.write( "F(X,Y) = X*Y + X*Y' + X'Y .... distributive law\r<BR>\n" );
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document.write( "At this point, we have the boolean function written as the sum as these specific minterms<BR>\n" );
document.write( "m3 = XY<BR>\n" );
document.write( "m2 = XY'<BR>\n" );
document.write( "m1 = X'Y<BR>\n" );
document.write( "If we wanted, we could express that sum as F(X,Y) = m1+m2+m3 = Σ(1,2,3)\r<BR>\n" );
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document.write( "This is the blank K-map for the two variables X and Y.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/yooV4Uo.png\"><BR>\n" );
document.write( "Along the top we have 0 and 1 corresponding to either X' or X respectively.<BR>\n" );
document.write( "Along the left side we have 0 and 1 corresponding to either Y' or Y respectively.<BR>\n" );
document.write( "These rules apply to minterms only.\r<BR>\n" );
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document.write( "Inside the table itself, we'll enter the m0 through m3 as shown below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/flzy1dN.png\"><BR>\n" );
document.write( "Here's some informal notation to help keep track.<BR>\n" );
document.write( "m0 = 00 = X'Y'<BR>\n" );
document.write( "m1 = 01 = X'Y<BR>\n" );
document.write( "m2 = 10 = XY'<BR>\n" );
document.write( "m3 = 11 = XY<BR>\n" );
document.write( "Or you could use the minterm table mentioned at the top of the page.\r<BR>\n" );
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document.write( "Next, we replace m1,m2,m3 with 1's since they are part of the F(X,Y) function after we expanded things out.<BR>\n" );
document.write( "The fact that m0 is left out means it gets assigned 0.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/dN7phTw.png\">\r<BR>\n" );
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document.write( "Now comes the grouping. <BR>\n" );
document.write( "Use these K-Map grouping rules<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.ee.surrey.ac.uk/Projects/Labview/minimisation/karrules.html\">http://www.ee.surrey.ac.uk/Projects/Labview/minimisation/karrules.html</a><BR>\n" );
document.write( "and you can review on this page if needed<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.ee.surrey.ac.uk/Projects/Labview/minimisation/karnaugh.html#introduction\">http://www.ee.surrey.ac.uk/Projects/Labview/minimisation/karnaugh.html#introduction</a><BR>\n" );
document.write( "There are a lot of rules to keep in mind, but once you get the hang of it, it's not too bad.\r<BR>\n" );
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document.write( "The goal is to group the '1's together in batches of powers of 2. In this case, it means we either want groups of 1 or groups of 2.<BR>\n" );
document.write( "The other goal is to minimize the number of groups, while also maximizing the group sizes.\r<BR>\n" );
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document.write( "Doing both of those tasks can be achieved with this grouping here.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/rgXGj7M.png\"><BR>\n" );
document.write( "Groups can overlap.<BR>\n" );
document.write( "The blue group represents X because XY'+XY = X(Y'+Y) = X*1 = X<BR>\n" );
document.write( "Similarly, the green group represents Y<BR>\n" );
document.write( "X'Y+XY = (X'+X)Y = 1*Y = Y<BR>\n" );
document.write( "Therefore, we end up with <font color=red>X+Y</font> as the final simplification.\r<BR>\n" );
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document.write( "Here's the boolean verification table<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >Y</TD><TD >X’</TD><TD >X’Y</TD><TD >X+X’Y</TD><TD >X+Y</TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD></tr><tr><TD >0</TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >1</TD></tr><tr><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >1</TD><TD >1</TD></tr><tr><TD >1</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >1</TD><TD >1</TD></tr>\n" );
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document.write( "The last two columns for X+X'Y and X+Y are the same.<BR>\n" );
document.write( "This proves that X+X'Y = X+Y\r<BR>\n" );
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document.write( "Answer: <font color=red>X+Y</font><BR>\n" );
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