document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191166>Question 1191166</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify the following functions using a K-map<BR>\n" );
document.write( "a.  F(X,Y, Z) = π ( 0, 2, 5,7)<BR>\n" );
document.write( "b.  F(X, Y, Z) =  XY’Z + X’ + Z + Y’Z’<BR>\n" );
document.write( "c.  F(W, X, Y, Z) = X’ Y’Z’ +  XYZ’ + WXY + W’X’Y’ + WZ<BR>\n" );
document.write( "d.  F(W, X, Y, Z) = X’  +  XZ’ + WX’Y + W’Y’ + WZ\r<BR>\n" );
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document.write( "Can you also explain me this second homework step-by-step please? 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 #823090 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=823090\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll do part A to get you started. \r<BR>\n" );
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document.write( "If you still need help with the others, then please make a new post. <BR>\n" );
document.write( "Please post <u>one problem</u> at a time. <BR>\n" );
document.write( "The solution page tends to get a bit cluttered when there are multiple problems per post. <BR>\n" );
document.write( "Though to be fair, my responses get a bit lengthy/wordy sometimes. I apologize in advance.\r<BR>\n" );
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document.write( "========================================================================================================\r<BR>\n" );
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document.write( "Part A\r<BR>\n" );
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document.write( "Maxterms (3 variables)<BR>\n" );
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document.write( "<tr><TD  colspan=3></TD><TD  colspan=2> &nbsp; &nbsp;&nbsp; Maxterms</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+Y'+Z</TD><TD >M2</TD></tr><tr><TD >0</TD><TD >1</TD><TD >1</TD><TD >X+Y'+Z'</TD><TD >M3</TD></tr><tr><TD >1</TD><TD >0</TD><TD >0</TD><TD >X'+Y+Z</TD><TD >M4</TD></tr><tr><TD >1</TD><TD >0</TD><TD >1</TD><TD >X'+Y+Z'</TD><TD >M5</TD></tr><tr><TD >1</TD><TD >1</TD><TD >0</TD><TD >X'+Y'+Z</TD><TD >M6</TD></tr><tr><TD >1</TD><TD >1</TD><TD >1</TD><TD >X'+Y'+Z'</TD><TD >M7</TD></tr>\n" );
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document.write( "F(X, Y, Z) = π ( 0, 2, 5,7) ..... product pi function of those listed maxterms<BR>\n" );
document.write( "F(X, Y, Z) = M0 * M2 * M5 * M7 .... another way to state the product<BR>\n" );
document.write( "F(X, Y, Z) = (M0) * (M2) * (M5) * (M7)<BR>\n" );
document.write( "F(X, Y, Z) = (X+Y+Z) * (X+Y'+Z) * (X'+Y+Z') * (X'+Y'+Z') .... use the table above\r<BR>\n" );
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document.write( "Let's set up the K-map, more formally known as the Karnaugh Map, but I'll refer to it as K-map from now on.\r<BR>\n" );
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document.write( "We have n = 3 literals X,Y,Z.<BR>\n" );
document.write( "This yields 2^n = 2^3 = 8 different inner cells of the K-map.<BR>\n" );
document.write( "This particular K-map is going to be a table with 2 rows and 4 columns to produce the 2*4 = 8 cells needed.\r<BR>\n" );
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document.write( "Along the left side, we'll have 0 and 1 to represent the idea of either X or X' in that order. <BR>\n" );
document.write( "This applies to maxterms only. <BR>\n" );
document.write( "With minterms, the order would be swapped (0 goes with X' and 1 goes with X).\r<BR>\n" );
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document.write( "Along the top we have these values: 00, 01, 11, 10<BR>\n" );
document.write( "The order is important.<BR>\n" );
document.write( "The first slot is the Y position, and the second slot is the Z position.<BR>\n" );
document.write( "0 in the 1st slot means Y, 1 in the 1st slot means Y'<BR>\n" );
document.write( "0 in the 2nd slot means Z, 1 in the 2nd slot means Z'\r<BR>\n" );
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document.write( "You might be thinking \"wait a minute, isn't 11 supposed to be AFTER 10 in binary?\". <BR>\n" );
document.write( "You'd be correct in thinking that, but I'm instead using Gray Code notation. <BR>\n" );
document.write( "More on that is provided at the following links for further reading.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.tutorialspoint.com/what-is-gray-code\">https://www.tutorialspoint.com/what-is-gray-code</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.learnabout-electronics.org/Digital/dig16.php#gray\">https://www.learnabout-electronics.org/Digital/dig16.php#gray</a><BR>\n" );
document.write( "Also, this link<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.learnabout-electronics.org/Digital/dig24.php\">https://www.learnabout-electronics.org/Digital/dig24.php</a><BR>\n" );
document.write( "makes a mention of a gray code sequence as well.\r<BR>\n" );
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document.write( "Basically when going from 01 to 11, we only change <u><font color=red>one binary digit</font></u>. <BR>\n" );
document.write( "Namely, we change the left digit from 0 to 1. Then going from 11 to 10 we flip the right-most bit. <BR>\n" );
document.write( "Once again, we change one digit at a time.<BR>\n" );
document.write( "Notice that when conventionally counting in binary we have 01 turn into 10 where both bits flip, which is what we don't want.<BR>\n" );
document.write( "I'm probably not explaining this very well, but hopefully those links will do a better job to clear up any confusion. \r<BR>\n" );
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document.write( "To summarize the table so far, we have:<ul><li>0 and 1 as a column on the left most side to represent either X or X' in that order.</li><li>00, 01, 11, 10 along the top to represent The combos of Y, Y', Z, Z'.</li></ul>Figure 1 shows what we have so far.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/hs2MhM6.png\"><BR>\n" );
document.write( "Only the left and top headers are filled in. The actual inner stuff is blank for now.\r<BR>\n" );
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document.write( "Now to fill in this table with an intermediate set of expressions. Those expressions being M0 through M7. <BR>\n" );
document.write( "We'll combine the headers we made to pinpoint the location of each maxterm.<BR>\n" );
document.write( "In the top left corner we have X = 0 and YZ = 00. They combine to XYZ = 000. I'm not multiplying these items. I'm doing string concatenation. My use of notation here is probably a bit unconventional but hopefully you get the idea of what I mean.\r<BR>\n" );
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document.write( "Refer to the maxterms chart above to see that (X,Y,Z) = (0,0,0) refers to M0.<BR>\n" );
document.write( "As another example, the bottom right corner has X = 1 and YZ = 10 to produce XYZ = 110. This binary number 110 corresponds to M6.<BR>\n" );
document.write( "The other M values will follow the same idea.<BR>\n" );
document.write( "Once all the M0 through M7 values are in place, you should get what you see in Figure 2 as shown above.\r<BR>\n" );
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document.write( "Next we replace the M values with either 0's or 1's.<BR>\n" );
document.write( "More specifically we replace the following M terms {M0,M2,M5,M7} with 0s. Everything else gets a 1.\r<BR>\n" );
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document.write( "Check out Figure 3 to see what the K-map should look like once it is filled out completely (minus any groupings just yet).<BR>\n" );
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document.write( "Then in figure 4, I've color-coded the groups as described by the rules/steps mentioned in these links<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.learnabout-electronics.org/Digital/dig24.php\">https://www.learnabout-electronics.org/Digital/dig24.php</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.javatpoint.com/simplification-of-boolean-expressions-using-karnaugh-map\">https://www.javatpoint.com/simplification-of-boolean-expressions-using-karnaugh-map</a><BR>\n" );
document.write( "There are a few rules to keep in mind, but the idea is to group up the 0s together (this applies to maxterms only).<BR>\n" );
document.write( "Each group of zeros has a count of either 1, 2, 4, 8, 16, ... ie some power of 2.<BR>\n" );
document.write( "We can only group to the immediate adjacent neighbor but not along any diagonal. <BR>\n" );
document.write( "We can span off the edge and end up on the other side to form a group that way, which is how I formed group A.<BR>\n" );
document.write( "The first link goes into more detail why we can join up the edges. The link shows a cylinder shape in one of their diagrams to illustrate what's going on.<BR>\n" );
document.write( "The goal of simplification is of course to have as few groups as possible.\r<BR>\n" );
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document.write( "Now to figure out what to do with the groups (you'll have to pretend that the very bottom of figure 4 didn't spoil the punchline).<BR>\n" );
document.write( "In group A, we have X = 0 and Z = 0. This produces the sum X+Z. We use zeros for maxterms for the original literal.<BR>\n" );
document.write( "In group B, we have X = 1 and Z = 1. This produces the sum X'+Z'. Ones indicate complements of the original literal.<BR>\n" );
document.write( "Therefore, the product of these groups is <font color=red>F(X, Y, Z) = (X + Z)(X' + Z') which is the final answer.</font>\r<BR>\n" );
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document.write( "So we can say F(X, Y, Z) = π( 0, 2, 5, 7) = <font color=red>(X + Z)(X' + Z')</font><BR>\n" );
document.write( "Or more thoroughly,<BR>\n" );
document.write( "π( 0, 2, 5, 7) = (X+Y+Z) * (X+Y'+Z) * (X'+Y+Z') * (X'+Y'+Z') = <font color=red>(X + Z)(X' + Z')</font>\r<BR>\n" );
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document.write( "Notice how Y is nowhere to be found in the simplified product-of-sums answer. <BR>\n" );
document.write( "This is because neither the presence of Y nor Y' affect the final outcome based on what the K-map shows. <BR>\n" );
document.write( "In the first pink cell of group A (on the left) we have Y = 0; in the right pink cell we have Y = 1. <BR>\n" );
document.write( "For both cells mentioned, the result is 0. Hence this is the reason the literal Y doesn't matter, so it can be eliminated.\r<BR>\n" );
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document.write( "For similar reasoning, group B doesn't have Y either. The Y is entirely redundant.<BR>\n" );
document.write( "If your teacher has introduced circuit analysis, try to draw a circuit diagram so you can spot the visual reason/proof the variable Y is not needed. This is an exercise left to the reader, but feel free to ask if you get stuck. \r<BR>\n" );
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document.write( "Here is a useful K-map calculator to check your work.<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" );
document.write( "Try not to rely solely on the calculator without having to do any (home)work or thinking beforehand.<BR>\n" );
document.write( "Rather, you should only use it when to check your work or perhaps use it when you get stuck somehow.<BR>\n" );
document.write( "There are other handy free online calculators as well you can search out, if you prefer.\r<BR>\n" );
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document.write( "It appears that calculator only accepts minterms (well the index number of them anyway). So you'll have to keep that in mind for maxterm problems.<BR>\n" );
document.write( "Instead of typing in the list of maxterms 0,2,5,7 you would type in the minterms 1,3,4,6. <BR>\n" );
document.write( "The calculator has a nice feature to swap between Product-of-Sums and Sum-of-Products formats.<BR>\n" );
document.write( "It displays the algebraic notation, as well as of course the Karnaugh Map itself when you scroll down to the bottom of the page.<BR>\n" );
document.write( "The map also shows how the terms group up (one marked in pink, the other in light blue).<BR>\n" );
document.write( "As you swap between Product-of-Sums and Sum-of-Products formats, the groupings will change but the numbers themselves will stay fixed.<BR>\n" );
document.write( "This in my opinion is a very nice way to tie minterms and maxterms together in a visual sense.\r<BR>\n" );
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document.write( "Here's how you could simplify without the use of a K-map. This is an optional section.\r<BR>\n" );
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document.write( "First we'll need a handy reference of Boolean Algebra laws (unless you've memorized them and don't need the table)<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Name</TD><TD >AND form</TD><TD >OR form</TD></tr><tr><TD >Identity Law</TD><TD >1*A = A</TD><TD >A+0 = A</TD></tr><tr><TD >Null Law</TD><TD >0*A = 0</TD><TD >A+1 = 1</TD></tr><tr><TD >Idempotent Law</TD><TD >A*A = A</TD><TD >A+A = A</TD></tr><tr><TD >Inverse Law</TD><TD >A*A' = 0</TD><TD >A+A' = 1</TD></tr><tr><TD >Commutative Law</TD><TD >A*B = B*A</TD><TD >A+B = B+A</TD></tr><tr><TD >Associative Law</TD><TD >A*(B*C) = (A*B)*C</TD><TD >A+(B+C) = (A+B)+C</TD></tr><tr><TD >Distributive Law</TD><TD >A+B*C = (A+B)*(A+C)</TD><TD >A*(B+C) = A*B+A*C</TD></tr><tr><TD >Absorption Law</TD><TD >A*(A+B) = A</TD><TD >A+A*B = A</TD></tr><tr><TD >De Morgan's Law</TD><TD >(A*B)' = A' + B'</TD><TD >(A+B)' = A'*B'</TD></tr>\n" );
document.write( "</TABLE>The star or asterisk symbol is sometimes omitted.<BR>\n" );
document.write( "Overhead horizontal bars are sometimes used in place of tickmarks to mean complements.\r<BR>\n" );
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document.write( "I'll also be using this standard list of minterms (3 variables)<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
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( "From there we can say this:<BR>\n" );
document.write( "F(X, Y, Z) = π ( 0, 2, 5, 7)  ............... product of given maxterms<BR>\n" );
document.write( "F(X, Y, Z) = ∑ ( 1, 3, 4, 6) ................ sum of minterms not previously listed<BR>\n" );
document.write( "F(X, Y, Z) = m1 + m3 + m4 + m6 ............. translate from sigma form to an actual sum of minterms<BR>\n" );
document.write( "F(X, Y, Z) = X'Y'Z + X'YZ + XY'Z' + XYZ' ..... replace each minterm using the table above<BR>\n" );
document.write( "F(X, Y, Z) = X'(Y'Z + YZ) + X(Y'Z' + YZ') ....... Distributive Law<BR>\n" );
document.write( "F(X, Y, Z) = X'((Y'+Y)Z) + X((Y'+Y)Z') ....... Distributive Law<BR>\n" );
document.write( "F(X, Y, Z) = X'((1)Z) + X((1)Z')  ....... Inverse Law<BR>\n" );
document.write( "F(X, Y, Z) = X'Z + XZ' ....... Identity Law.<BR>\n" );
document.write( "F(X, Y, Z) = XZ' + X'Z ....... Commutative Law<BR>\n" );
document.write( "F(X, Y, Z) = 0+X*Z'+Z*X'+0  ....... Identity Law<BR>\n" );
document.write( "F(X, Y, Z) = X*X'+X*Z'+Z*X'+Z*Z' ....... Inverse Law<BR>\n" );
document.write( "F(X, Y, Z) = X(X'+Z')+Z(X'+Z') ....... Distributive Law<BR>\n" );
document.write( "<font color=red>F(X, Y, Z) = (X + Z)(X' + Z')</font>  ....... Distributive Law\r<BR>\n" );
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document.write( "In step 2, I converted to a sum of products because I think that's the easier route to take. <BR>\n" );
document.write( "Though of course you might find the product-of-sums method easier.\r<BR>\n" );
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document.write( "Product-of-Sums answer format:<BR>\n" );
document.write( "<font color=red>F(X, Y, Z) = (X + Z)(X' + Z')</font>\r<BR>\n" );
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document.write( "Sum-of-Products answer format:<BR>\n" );
document.write( "F(X, Y, Z) = X'Z + XZ'\r<BR>\n" );
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document.write( "Since you started with the product pi function, it probably makes sense to stick with the product-of-sums answer format.<BR>\n" );
document.write( "Though it's beneficial to know how to go back and forth between each form.\r<BR>\n" );
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document.write( "Here is the verification boolean table (which may be optional based on your teacher's requirements).<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://docs.google.com/spreadsheets/d/1dfZ5jU_fWKpOaWb1WYjyAU2I1ykmDqZTiyj81yLNlVo/edit?usp=sharing\">https://docs.google.com/spreadsheets/d/1dfZ5jU_fWKpOaWb1WYjyAU2I1ykmDqZTiyj81yLNlVo/edit?usp=sharing</a><BR>\n" );
document.write( "You don't need to have a google account to be able to view the page.\r<BR>\n" );
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document.write( "Answer to part A is <font color=red>F(X, Y, Z) = (X + Z)(X' + Z')</font><BR>\n" );
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