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Revision as of 14:04, 2 April 2009
, 14:04, 2 April 2009→Comment : Sets as Logical Sums
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: (1 + ''a'')+: (1 + ''b'')+: (1 + ''c'')+<pre>+: 1 + ''a'' + ''b'' + ''c'' + ''ab'' + ''ac'' + ''bc'' + ''abc''.+: {}, {''a''}, {''b''}, {''c''}, {''a'', ''b''}, {''a'', ''c''}, {''b'', ''c''}, {''a'', ''b'', ''c''}.+
Probably on account of all those years I flippered away playing the oldtime pinball machines, I tend to imagine a product like this being displayed in a vertical array:
Probably on account of all those years I flippered away playing the oldtime pinball machines, I tend to imagine a product like this being displayed in a vertical array:
−{| align="center" cellspacing="6" width="90%"
−|
−<math>\begin{matrix}
+(1 ~+~ a)
+\\
+(1 ~+~ b)
+\\
+(1 ~+~ c)
+\end{matrix}</math>
+|}
−I picture this as a playboard with six "bumpers", the ball chuting down the board in such a career that it strikes exactly one of the two bumpers on each and every one of the three levels.
+I picture this as a playboard with six bumpers, the ball chuting down the board in such a career that it strikes exactly one of the two bumpers on each and every one of the three levels.
−So a trajectory of the ball where it hits the <math>a\!</math> bumper on the 1st level, hits the <math>1\!</math> bumper on the 2nd level, hits the <math>c\!</math> bumper on the 3rd level, and then exits the board, represents a single term in the desired product and corresponds to the subset <math>\{ a, c \}.\!</math>
−So a trajectory of the ball where it
−hits the "a" bumper on the 1st level,
−hits the "1" bumper on the 2nd level,
−hits the "c" bumper on the 3rd level,
−and then exits the board, represents
−a single term in the desired product
−and corresponds to the subset {a, c}.
−</pre>
−Multiplying out (1 + ''a'')(1 + ''b'')(1 + ''c''), one obtains:
+Multiplying out the product <math>(1 + a)(1 + b)(1 + c)\!</math>, one obtains:
−{| align="center" cellspacing="6" width="90%"
+|
+<math>\begin{array}{*{15}{c}}
+1 & + & a & + & b & + & c & + & ab & + & ac & + & bc & + & abc
+\end{array}</math>
+|}
And this informs us that the subsets of choice are:
And this informs us that the subsets of choice are:
−{| align="center" cellspacing="6" width="90%"
+|
+<math>\begin{matrix}
+\varnothing, & \{ a \}, & \{ b \}, & \{ c \}, & \{ a, b \}, & \{ a, c \}, & \{ b, c \}, & \{ a, b, c \}
+\end{matrix}</math>
+|}
==Selection 7==
==Selection 7==
