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Using the coproduct operator (<math>\textstyle\coprod</math>) for this construction, the ''sum'', the ''coproduct'', or the ''disjointed union'' of <math>P\!</math> and <math>Q\!</math> in that order can be represented as the ordinary union of <math>P_{[1]}\!</math> and <math>Q_{[2]}.\!</math>

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

the ordinary disjoint union of P_[1] and Q_[2].

P ]_[ Q   =   P_[1] |_| Q_[2].

P \coprod Q & = & P_{[1]} \cup Q_{[2]}. \\

The concatenation L_1 · L_2 of the formal languages L_1 and L_2 is

The concatenation L_1 · L_2 of the formal languages L_1 and L_2 is

just the cartesian product of sets L_1 x L_2 without the extra x's,

just the cartesian product of sets L_1 x L_2 without the extra x's,