document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191165>Question 1191165</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify the following functions using a K-map<BR>\n" );
document.write( "a. F(X, Y) = m2 + m3<BR>\n" );
document.write( "b. F(X, Y) = X + X’Y<BR>\n" );
document.write( "c. F(X, Y) = X’ + XY’<BR>\n" );
document.write( "d. F(X, Y, Z) = m0 + m2 + m5 + m7<BR>\n" );
document.write( "e. 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 this homework question step-by-step? Thank you!<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849141 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849141\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the simplification of the given functions using K-maps:\r<BR>\n" );
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document.write( "**a. F(X, Y) = m2 + m3**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "      Y<BR>\n" );
document.write( "   X  0  1<BR>\n" );
document.write( "   0  0  1  (m2)<BR>\n" );
document.write( "   1  0  1  (m3)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Grouping m2 and m3, we get:\r<BR>\n" );
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document.write( "F(X, Y) = Y\r<BR>\n" );
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document.write( "**b. F(X, Y) = X + X’Y**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "      Y<BR>\n" );
document.write( "   X  0  1<BR>\n" );
document.write( "   0  1  1<BR>\n" );
document.write( "   1  1  1<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Grouping all the 1s, we get:\r<BR>\n" );
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document.write( "F(X, Y) = 1\r<BR>\n" );
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document.write( "**c. F(X, Y) = X’ + XY’**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "      Y<BR>\n" );
document.write( "   X  0  1<BR>\n" );
document.write( "   0  1  1<BR>\n" );
document.write( "   1  1  0<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Grouping the top row and the single 1 at X=1, Y=0 we get:\r<BR>\n" );
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document.write( "F(X, Y) = X' + Y'\r<BR>\n" );
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document.write( "**d. F(X, Y, Z) = m0 + m2 + m5 + m7**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "       YZ<BR>\n" );
document.write( "   X  00  01  11  10<BR>\n" );
document.write( "   0  1   0   1   1<BR>\n" );
document.write( "   1  0   1   1   0<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Grouping m0 and m2, and m5 and m7:\r<BR>\n" );
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document.write( "F(X, Y, Z) = X’Y’ + YZ\r<BR>\n" );
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document.write( "**e. F(X, Y,Z) = X’Y’Z’ + X’YZ + XY’Z + XYZ**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "       YZ<BR>\n" );
document.write( "   X  00  01  11  10<BR>\n" );
document.write( "   0  1   1   0   1<BR>\n" );
document.write( "   1  1   1   1   1<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Grouping the top row, and the bottom row, we get:\r<BR>\n" );
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document.write( "F(X, Y, Z) = Y' + Z<BR>\n" );
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