Boolean Algebra

Sets

“Naive set theory” uses a set of well-defined elements—its universe—and considers a series of relations and operations between those elements and subsets of this universe. For example, if the universe is the set of all national flags at some specific moment in time, one can easily imagine all kinds of subsets: flags using red, black, or both; those featuring a cross, a circle, bars, stripes; and so forth. Using red is a binary variable, with value YES (or 1) or NO (or 0) for each flag.

Although not absolutely needed for conceptual development, a visual representation using Venn diagrams can show this relationship (see Figure 2). This figure shows the flag Universe and three of its subsets, namely flags using red (R), those using black (B), and those using yellow (Y). It immediately can be seen in the diagram that each subset automatically determines its complement (flags without black, without red, or without yellow). The Italian flag (i), for example, belongs to subset R but to neither B nor Y; this can be noted as i ∊ R, i ∉ B, and i ∉ Y.

It is also possible to consider operations on subsets. The intersection (∩) of R and B is the subset of all flags using both red AND black; the union (∪) of R and B is the subset of all flags using either red OR black—or both; this is the inclusive variant of OR. One can see that the French flag (f) satisfies f ∊ R ∪ B and f ∉ R ∩ B. Finally, some subset may be totally included (⊂) within another: The set of all flags using both red and black is included in the set of all flags that merely use red. This is noted as R ∩ B ⊂ R.

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