Changes

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.

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.

For example, given the sets <math>P\!</math> and <math>Q,\!</math> overlapping or not, one can define the ''indexed'' or ''marked'' sets <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> amounting to the copy of <math>P\!</math> into the first part of the co-product and the copy of <math>Q\!</math> into the second part of the co-product, in the following manner:

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

copy of P into the first part of the co-product and the copy of Q into the

second part of the co-product, in the following manner:

P_[1] = <P, 1>  = {<x, 1>  : x in P},

 

Q_[2] = <Q, 2>  = {<x, 2>  : x in Q}.

P_{[1]} & = & (P, 1) & = & \{ (x, 1) : x \in P \}, \\

Q_{[2]} & = & (Q, 2) & = & \{ (x, 2) : x \in Q \}. \\

Using the sign "]_[" for this construction, the "sum", the "co-product",

Using the sign "]_[" for this construction, the "sum", the "co-product",

or the "disjointed union" of P and Q in that order can be represented as

or the "disjointed union" of P and Q in that order can be represented as