## Indicator Functions : 1

One of the things it helps to understand about 19th Century mathematicians, and those who built the bridge to the 20th, is that they were capable of high abstraction — in Peirce’s case a cut above what is common today — and yet they remained close enough to the point where abstract forms were teased away from the concrete materials of mathematical inquiry that they were able to maintain a sense of connection between the two.  There are few better places to see this connection than in the medium of Venn diagrams.  But Venn diagrams are such familiar pictures that it’s easy to overlook their subtleties, so it may be useful to spend some time developing the finer points of what they picture.

There are actually several types of Boolean functions that are represented in a typical Venn diagram.  They all have the Boolean domain $\mathbb{B} = \{ 0, 1 \}$ or one of its powers $\mathbb{B}^k = \{ 0, 1 \}^k$ as their functional codomains but their functional domains may vary, not all being limited to finite cardinalities.  As an aid to sorting out their variety, consider the array of functional arrows in the following figure.

Suppose $X$ is a universe of discourse represented by the rectangular area of a Venn diagram.  Note that the set $X$ itself may have any cardinality.  The most general type of Boolean function is a map $f : X \to \mathbb{B}.$  This is known as a Boolean-valued function since only its functional values need be in $\mathbb{B}.$

A function of the type $f : X \to \mathbb{B}$ is called a characteristic function in set theory or an indicator function in probability and statistics since it can be taken to characterize or indicate a particular subset $S$ of $X,$ namely, the fiber or inverse image of the value $1,$ for which we have the notation and definition $f^{-1}(1) = \{ x \in X : f(x) = 1 \}.$

The notation $f_S$ is often used for the characteristic function of a subset $S$ of $X.$  Putting all the pieces together then, we have $f_S^{-1}(1) = S \subseteq X.$

To be continued …

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