document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209457>Question 1209457</A>: <I><SPAN STYLE=\"word-break: break-all;\"> List all subsets of {A,B,C,D} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848872 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=848872\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "A set is a collection of things. In this case letters A through D.<BR>\n" );
document.write( "A subset is a smaller set that contains some of those letters. <BR>\n" );
document.write( "It could be 0 of them, which gives the empty set. The empty set is a subset of every set.<BR>\n" );
document.write( "It could be all of them. Any set is a subset of itself. <BR>\n" );
document.write( "Or it could be something like {A,B,C} or {B,D}\r<BR>\n" );
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document.write( "To systematically list out every possible subset, we can use binary notation.<BR>\n" );
document.write( "We count from 0 to (2^4)-1 = 15 in binary. The exponent 4 refers to the number of elements in the original set.<BR>\n" );
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document.write( "<tr><TD >Base10</TD><TD >Base2</TD></tr><tr><TD >0</TD><TD >0000</TD></tr><tr><TD >1</TD><TD >0001</TD></tr><tr><TD >2</TD><TD >0010</TD></tr><tr><TD >3</TD><TD >0011</TD></tr><tr><TD >4</TD><TD >0100</TD></tr><tr><TD >5</TD><TD >0101</TD></tr><tr><TD >6</TD><TD >0110</TD></tr><tr><TD >7</TD><TD >0111</TD></tr><tr><TD >8</TD><TD >1000</TD></tr><tr><TD >9</TD><TD >1001</TD></tr><tr><TD >10</TD><TD >1010</TD></tr><tr><TD >11</TD><TD >1011</TD></tr><tr><TD >12</TD><TD >1100</TD></tr><tr><TD >13</TD><TD >1101</TD></tr><tr><TD >14</TD><TD >1110</TD></tr><tr><TD >15</TD><TD >1111</TD></tr>\n" );
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document.write( "Counting in binary shouldn't be too tricky once you get a hang of it. <BR>\n" );
document.write( "Basically you start at 0000 and increment the last digit by 1 to get 0001<BR>\n" );
document.write( "Then increment the last digit of 0001 to get 0002, but the \"2\" doesn't exist in binary. <BR>\n" );
document.write( "The only digits are 0 and 1. <BR>\n" );
document.write( "A fundamental rule in binary is 1<sub>2</sub>+1<sub>2</sub> = 10<sub>2</sub> or you can phrase it as saying \"1+1 = 10 in binary\".<BR>\n" );
document.write( "This explains how we go from 0001 to 0010. The 1 carries over to the left.<BR>\n" );
document.write( "Then we go from 0010 to 0011<BR>\n" );
document.write( "Then from 0011 to 0100<BR>\n" );
document.write( "And so on.<BR>\n" );
document.write( "A recommended tool is to use the <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/dec2bin-function-0f63dd0e-5d1a-42d8-b511-5bf5c6d43838\">Dec2Bin</a> spreadsheet function.<BR>\n" );
document.write( "The first input of this function is the number we want to convert to binary. The second input is the number of digits for the output. <BR>\n" );
document.write( "For example type in <font color=red>=Dec2Bin(A1,4)</font> to convert whatever is in cell A1 to binary. It will have 4 binary digits. <BR>\n" );
document.write( "There are many online calculators that can convert from decimal (base 10) to binary (base 2).\r<BR>\n" );
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document.write( "Why go through the trouble of using binary? Because 0 represents leaving out a particular letter while 1 represents including it. <BR>\n" );
document.write( "String those 1's and/or 0's together to form a list of items. <BR>\n" );
document.write( "0101 for instance means we exclude A, include B, exclude C, include D.<BR>\n" );
document.write( "In short basically you have B and D to form the subset {B,D}\r<BR>\n" );
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document.write( "Another example is 1110 represents the subset {A,B,C} since we have 1's as the first three slots and 0 in the last slot.<BR>\n" );
document.write( "As you can see this guarantees we do not overlook any subset. \r<BR>\n" );
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document.write( "Let's introduce a third column to the previous table that lists out all the subsets possible.<BR>\n" );
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document.write( "<tr><TD >Base10</TD><TD >Base2</TD><TD >Subset</TD></tr><tr><TD >0</TD><TD >0000</TD><TD >{  }</TD></tr><tr><TD >1</TD><TD >0001</TD><TD >{ D }</TD></tr><tr><TD >2</TD><TD >0010</TD><TD >{ C }</TD></tr><tr><TD >3</TD><TD >0011</TD><TD >{C, D}</TD></tr><tr><TD >4</TD><TD >0100</TD><TD >{ B }</TD></tr><tr><TD >5</TD><TD >0101</TD><TD >{ B, D }</TD></tr><tr><TD >6</TD><TD >0110</TD><TD >{ B, C }</TD></tr><tr><TD >7</TD><TD >0111</TD><TD >{ B, C, D }</TD></tr><tr><TD >8</TD><TD >1000</TD><TD >{ A }</TD></tr><tr><TD >9</TD><TD >1001</TD><TD >{ A, D }</TD></tr><tr><TD >10</TD><TD >1010</TD><TD >{ A, C }</TD></tr><tr><TD >11</TD><TD >1011</TD><TD >{ A, C, D}</TD></tr><tr><TD >12</TD><TD >1100</TD><TD >{ A, B }</TD></tr><tr><TD >13</TD><TD >1101</TD><TD >{ A, B, D }</TD></tr><tr><TD >14</TD><TD >1110</TD><TD >{ A, B, C }</TD></tr><tr><TD >15</TD><TD >1111</TD><TD >{ A, B, C, D }</TD></tr>\n" );
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document.write( "Optionally we can sort the subsets like so<BR>\n" );
document.write( "{   }<BR>\n" );
document.write( "{ A }<BR>\n" );
document.write( "{ B }<BR>\n" );
document.write( "{ C }<BR>\n" );
document.write( "{ D }<BR>\n" );
document.write( "{ A, B }<BR>\n" );
document.write( "{ A, C }<BR>\n" );
document.write( "{ B, C }<BR>\n" );
document.write( "{ A, D }<BR>\n" );
document.write( "{ B, D }<BR>\n" );
document.write( "{ C, D }<BR>\n" );
document.write( "{ A, B, C }<BR>\n" );
document.write( "{ A, B, D }<BR>\n" );
document.write( "{ A, C, D }<BR>\n" );
document.write( "{ B, C, D }<BR>\n" );
document.write( "{ A, B, C, D }<BR>\n" );
document.write( "Feel free to sort them any way you prefer.\r<BR>\n" );
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document.write( "{ } is the empty set. It has nothing inside it. Not even 0 is inside. The empty set is a subset of every set. Sometimes a special symbol is used for the empty set. The symbol looks like a zero with a slash through it.\r<BR>\n" );
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document.write( "The <a  rel=nofollow HREF=\"https://mathworld.wolfram.com/SingletonSet.html\">singletons</a> are listed next. Singletons have 1 element as you can probably guess by the name. Then after the singletons are subsets with 2 elements. Then 3 element subsets and so on.<BR>\n" );
document.write( "Since we have n = 4 slots and 2 choices per slot (either include the element or not), there are 2^n = 2^4 = 16 different subsets. The collection of all subsets is known as the <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Power_set\">Power Set</a>.<BR>\n" );
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