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The relation between logical disjunction and set-theoretic union and the relation between logical conjunction and set-theoretic intersection ought to be clear enough for the purposes of the immediately present context. In any case, all of these relations are scheduled to receive a thorough examination in a subsequent discussion (Subsection 1.3.10.13). But the relation of a set-theoretic union to a category-theoretic co-product and the relation of a set-theoretic intersection to a syntactic concatenation deserve a closer look at this point.
The relation between logical disjunction and set-theoretic union and the relation between logical conjunction and set-theoretic intersection ought to be clear enough for the purposes of the immediately present context. In any case, all of these relations are scheduled to receive a thorough examination in a subsequent discussion (Subsection 1.3.10.13). But the relation of a set-theoretic union to a category-theoretic co-product and the relation of a set-theoretic intersection to a syntactic concatenation deserve a closer look at this point.
The effect of a co-product as a ''disjointed union'', in other words, that creates an object tantamount to a disjoint union of sets in the resulting co-product even if some of these sets intersect non-trivially and even if some of them are identical ''in reality'', can be achieved in several ways. The most usual conception is that of making a ''separate copy'', for each part of the intended co-product, of the set that is intended to go there. Often one thinks of the set that is assigned to a particular part of the co-product as being distinguished by a particular ''color'', in other words, by the attachment of a distinct ''index'', ''label'', or ''tag'', being a marker that is inherited by and passed on to every element of the set in that part. A concrete image of this construction can be achieved by imagining that each set and each element of each set is placed in an ordered pair with the sign of its color, index, label, or tag. One describes this as the ''injection'' of each set into the corresponding ''part'' of the co-product.
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For example, given the sets P and Q, overlapping or not, one can define
For example, given the sets P and Q, overlapping or not, one can define
the "indexed" sets or the "marked" sets P_[1] and Q_[2], amounting to the
the "indexed" sets or the "marked" sets P_[1] and Q_[2], amounting to the
