Persephone would certainly know the Merovingian from their respective functions: the Merovingian (with his vision that resulted from being an operating system) was trying to help the Architect figure out how to solve the failing Matrix system, and Persephone was probably working with the Merovingian to aid in his understanding of how to manipulate humans into doing what the system wants. Love would surely have been the first logical choice to explore, and that is exactly what Persephone continues to learn about. Every chance she gets, she feeds off of the love of males... and females (as seen in Enter the Matrix).
It is probably largely due to the efforts of Persephone that machines now seem to fully understand love. Rama-Kandra feels love for his daughter, and he can read Neo's love for Trinity like a book - it seems to be "written all over" Neo's code in the Train Station. Even the Architect recognizes and understands what makes up love when he sees Neo's logic being overwhelmed by love.
When Persephone leads Morpheus, Neo and Trinity to the Keymaker, she does so to get even with the Merovingian. While it may seem extreme to cause the Merovingian to lose all of his bodyguards just because the Merovingian cheated on her in the womans' bathroom, we have to remember that love is just as important to Persephone as power is to the Merovingian. The Merovingian tells Persephone with regard to love, "It's a game. It's only a game," which shows disrespect toward the love that is important to her. Persephone, regarding the Merovingian's hold on power, says, "So is this. Have fun," showing equal disrespect toward the power that is important to him.
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