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Selections and set theory

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It is possible to think of sets as selections rather than mere 'collections.' But what does this mean? To see how this works, construct a chart with 3 columns and 3 rows. At the top of the first column put A and at the top of the second column put B. We have two sets from which we can select sets, A and B. Next, in the second row of the third column, put {A, B}. Now, it is possible to make selections from columns A and B. We have the options "neither A nor B; A but not B; B but not A; and both A and B"(Steinhart, p. 15). In order to denote which selections are being made, we place either 1 or 0 under the respective column. So for example, if you want your selection to be "both A and B," input "1" under both columns. Steinhart notes that this way of organizing selection options was developed by Neumann in 1923.

This signifies that a selection is being made from both columns. If you want "A but not B," input 1 under column A and 0 in column B. This signifies that you have selected A but not B. In order to signify B but not A, input a 0 in column A and a 1 but in column B. If you want neither, input 0 in both columns. Under the 3rd column, we can signify these selections with {A, B}, {A}, {B} and {}, respectively. Those who have studied logic will be familiar with this notation. 1 means "true" and 2 means "false." "Note that if there are n objects, then there are 2^n ways to select those objects. This is familiar with logic: if there are n propositions, there are 2^n ways to assign truth-values to those propositions"(Steinhart, p. 15). We have at our disposal six "objects": A, B, {}, {B}, {A}, {A, B}. Since we have two variables and 6 objects, there are 2^6 ways to select these objects. 2^6 = 64.

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.

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