Daugther, Mother, Child...

Claribel was sent to be married to the King of Tunsia by her father's orders. It was seen as a way to expand their family's reach to Africa. This would be good for their family. Even though the Africans were seen as a people of "wild" background, completely unrefined comparitvely to how Claribel lived in Europe. Daughter of a King she would of course be well read and educated to a level above her subjects in Africa. Because Claribel weds the King of Tunis, her father will now be able to benefit from the union as well.

Sycorax, is a an unfortunate soul that arrives at the island because she is exiled from Algeria. Her eviction from her home is likely one of shame due to the pregnancy. This is very typical of a woman that not virtuous and allows herself to fall to the passions of a man whom she is not betrothed to or promised to wed.  This makes me think that Sycorax came from a home where the values weren't instilled in her to wait for the proper man. Rather than her family embrace her they are shamed and send her away to what could well have been her death.

 
When Miranda sees Ferdinand she is in love with him nearly immediately. Her father begs of her to please understand that the world is not merely made up of monstrous beings like Caliban and the attractive Ferdinand. Prospero watches over his daughter Miranda when she is in contact with Ferdinand most likely because he wants her wed to someone with promise. Ferdinand is the furture King of Naples and therefore a perfect mate for his only child. However he cannot help but ensure his daughter's virtues remain in tact, by saying:

"Then, as my gift and thine own acquisition

Worthily purchased take my daughter: but

If thou dost break her virgin-knot before

All sanctimonious ceremonies may

With full and holy rite be minister'd,

No sweet aspersion shall the heavens let fall

To make this contract grow: but barren hate,

Sour-eyed disdain and discord shall bestrew

The union of your bed with weeds so loathly

That you shall hate it both: therefore take heed,

As Hymen's lamps shall light you." (4.i.14-24)