document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190973>Question 1190973</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify the following functions using Boolean Algebra<BR>\n" );
document.write( "a.  F(X, Y, Z) = X Y + X’ Y + X Z<BR>\n" );
document.write( "b.  F(X, Y, Z) = (X + Y) (X’ + Y + Z)<BR>\n" );
document.write( "c.  F(X, Y, Z) = X Y’ Z + X Y Z + Y’ Z<BR>\n" );
document.write( "d.  F(X, Y, Z) =    XY + X’YZ<BR>\n" );
document.write( "e.  F(X, Y, Z)  = X’Y + XYZ’<BR>\n" );
document.write( "f.  F(X, Y, Z)  =(XY) + (X + Y +Z)’X + YZ<BR>\n" );
document.write( "g.  F(X, Y, Z) =  (XY)’ + (X+Y +Z)’<BR>\n" );
document.write( "h.  F(X,Y,Z) =  YZ + (X+Y)’ +  (XYZ)’<BR>\n" );
document.write( "i.  F(X,Y,Z) =  (X+Y +Z)’ (X+Y)\r<BR>\n" );
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document.write( "Could you explain me this second homework question 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 #822746 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=822746\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We'll need these Boolean Algebra rules<BR>\n" );
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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( "This means something like A*B is the same as AB\r<BR>\n" );
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document.write( "In some Boolean algebra textbooks, the tickmark is replaced with a horizontal bar overhead.<BR>\n" );
document.write( "Eg: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large A' = \overline{A}\"><BR>\n" );
document.write( "I'll use the tickmark notation here.\r<BR>\n" );
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document.write( "Since you posted quite a lot of problems, I'll do the first 5 to get you started.<BR>\n" );
document.write( "I'll go from part (a) to part (e).\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( "X*Y + X'*Y + X*Z<BR>\n" );
document.write( "(X+X')*Y + X*Z ... Distributive Law<BR>\n" );
document.write( "(1)*Y + X*Z ... Inverse Law<BR>\n" );
document.write( "Y + X*Z ... Identity Law\r<BR>\n" );
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document.write( "Answer: Y + X*Z\r<BR>\n" );
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document.write( "Here's the verification table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >Y</TD><TD >Z</TD><TD >X’</TD><TD >X*Y</TD><TD >X’*Y</TD><TD >X*Z</TD><TD >X*Y+X’*Y+X*Z</TD><TD >Y + X*Z</TD></tr><tr><TD >1</TD><TD >1</TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >1</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 >0</TD><TD >0</TD><TD >1</TD><TD >1</TD></tr><tr><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</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 >0</TD><TD >0</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 >0</TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >1</TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >1</TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD></tr><tr><TD >0</TD><TD >0</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD></tr>\n" );
document.write( "</TABLE>The last two columns are identical, which means X*Y + X'*Y + X*Z is the same as Y + X*Z\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "(X+Y)*(X' +  Y + Z)<BR>\n" );
document.write( "W*(X' +  Y + Z) .... let W = X+Y<BR>\n" );
document.write( "W*X' + W*Y + W*Z .... Distributive Law<BR>\n" );
document.write( "X'*W + Y*W + Z*W .... Commutative Law<BR>\n" );
document.write( "X'*(X+Y) + Y*(X+Y) + Z*(X+Y) .... Replace every W with X+Y<BR>\n" );
document.write( "X'*X+X'*Y + Y*X+Y*Y + Z*X+Z*Y .... Distributive Law<BR>\n" );
document.write( "0+X'*Y + Y*X+Y*Y + Z*X+Z*Y .... Inverse Law<BR>\n" );
document.write( "X'*Y + Y*X+Y*Y + Z*X+Z*Y .... Identity Law<BR>\n" );
document.write( "X'*Y + Y*X+Y + Z*X+Z*Y .... Idempotent Law<BR>\n" );
document.write( "X'*Y + Y*X+Y+Z*Y + Z*X .... Commutative Law<BR>\n" );
document.write( "(X'+ X+1+Z)*Y + Z*X .... Distributive Law<BR>\n" );
document.write( "(1+1+Z)*Y + Z*X .... Inverse Law<BR>\n" );
document.write( "(1+Z)*Y + Z*X .... Idempotent Law<BR>\n" );
document.write( "1*Y + Z*Y + Z*X .... Distributive Law<BR>\n" );
document.write( "Y + Z*Y + Z*X .... Identity Law<BR>\n" );
document.write( "Y + Z*X .... Absorption Law\r<BR>\n" );
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document.write( "Answer: Y + Z*X\r<BR>\n" );
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document.write( "Side note: This is equivalent to the result of part (a)\r<BR>\n" );
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document.write( "Verification Table<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >Y</TD><TD >Z</TD><TD >X’</TD><TD >Y’</TD><TD >Z*X</TD><TD >X+Y</TD><TD >X’+Y+Z</TD><TD >(X+Y)*(X’+Y+Z)</TD><TD >Y + Z*X</TD></tr><tr><TD >1</TD><TD >1</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >1</TD></tr><tr><TD >1</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >1</TD></tr><tr><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >1</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 >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 >0</TD><TD >0</TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >1</TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >1</TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD></tr><tr><TD >0</TD><TD >0</TD><TD >0</TD><TD >1</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD></tr>\n" );
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document.write( "========================================================<BR>\n" );
document.write( "Part (c)\r<BR>\n" );
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document.write( "X*Y'*Z + X*Y*Z + Y'*Z<BR>\n" );
document.write( "X*Z*(Y' + Y) + Y'*Z .... Distributive Law<BR>\n" );
document.write( "X*Z*(1) + Y'*Z .... Inverse Law<BR>\n" );
document.write( "X*Z + Y'*Z .... Identity Law<BR>\n" );
document.write( "(X + Y')*Z ... Distributive Law\r<BR>\n" );
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document.write( "Answer: (X + Y')*Z\r<BR>\n" );
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document.write( "I'll let you create the verification table.\r<BR>\n" );
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document.write( "========================================================<BR>\n" );
document.write( "Part (d)\r<BR>\n" );
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document.write( "X*Y + X'*Y*Z<BR>\n" );
document.write( "(X*Y + X'*Y)*(X*Y + Z) ... Distributive Law (use the \"AND\" form)<BR>\n" );
document.write( "( (X + X')*Y )*(X*Y + Z) ... Distributive Law <BR>\n" );
document.write( "( (1)*Y )*(X*Y + Z) ... Inverse Law <BR>\n" );
document.write( "Y*(X*Y + Z) ... Identity Law <BR>\n" );
document.write( "Y*X*Y + Y*Z ... Distributive Law <BR>\n" );
document.write( "X*Y*Y + Y*Z ... Commutative Law <BR>\n" );
document.write( "X*Y + Y*Z ... Idempotent Law <BR>\n" );
document.write( "Y*(X+Z) ... Distributive Law \r<BR>\n" );
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document.write( "Answer: Y*(X+Z)\r<BR>\n" );
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document.write( "========================================================<BR>\n" );
document.write( "Part (e)\r<BR>\n" );
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document.write( "X'*Y + X*Y*Z'<BR>\n" );
document.write( "(X'*Y + X*Y)*(X'Y + Z') ... Distributive Law (use the \"AND\" form)<BR>\n" );
document.write( "( (X' + X)*Y)*(X'Y + Z') ... Distributive Law<BR>\n" );
document.write( "( (1)*Y)*(X'Y + Z') ... Inverse Law<BR>\n" );
document.write( "Y*(X'Y + Z') ... Identity Law<BR>\n" );
document.write( "Y*X'Y + Y*Z' ... Distributive Law<BR>\n" );
document.write( "X'*Y*Y + Y*Z' ... Commutative Law<BR>\n" );
document.write( "X'*Y + Y*Z' ... Idempotent Law<BR>\n" );
document.write( "Y*(X'+Z') ... Distributive Law<BR>\n" );
document.write( "Y*(X*Z)' ... De Morgan's Law\r<BR>\n" );
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document.write( "Answer: Y*(X*Z)'<BR>\n" );
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