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Is there any difference? What's their usage? When should one be used instead of the other one?

Examples:

El niño debe de hacer su tarea.

El niño debe hacer su tarea.

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By comparison with English: deber = must; deber de = should. –  pferor Feb 8 '12 at 20:23
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2 Answers

up vote 10 down vote accepted

The difference is simple:

  • Deber + infinitive is used to express obligation:

    You must do it as soon as possible.

  • Deber de + infinitive is used to express probability or supposition:

    It must be very early.

Sometimes, in the second case the preposition "de" is omitted, so it might be confusing. So the first case could also be interpretated as the second one, but the context of the sentence would give you the right meaning.

In your example I'd say that the correct way is "El niño debe hacer su tarea", because it's likely that the meaning is obligation. But for example you mean something like:

The test is answered, the child must have done his homework

then you would use "deber de" (but with another tense) because this is a supposition:

El test está contestado, el niño debe de haber hecho sus tareas.

You can find on the RAE the official reference about the topic.

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Thanks @dusan. I was going to correct it myself... :D –  Alenanno Dec 5 '11 at 17:14
    
@Alenanno No problem =) –  dusan Dec 5 '11 at 17:16
    
Here's the link to the DPD explanation about deber, in case you want to add it to your answer: buscon.rae.es/dpdI/SrvltConsulta?lema=deber –  Gonzalo Medina Dec 5 '11 at 19:43
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  1. El niño debe hacer su tarea.

    This sentence means obligation: The kid must do his job.

    A similar sentence could be "El niño tiene que hacer su tarea."

  2. El niño debe de hacer su tarea.

    The construction "deber de" means supposition or possibility. I'm not sure this usage applies to your example as is. But another example can be:

    Deben de ser las ocho y media. - It must be half past eight.

If you want to read more, check the "deber" vs "deber de" comparison on the RAE, provided by Gonzalo Medina.

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Here's the link to the DPD explanation about deber, in case you want to add it to your answer: buscon.rae.es/dpdI/SrvltConsulta?lema=deber. By the way, there's a little typo: "ocho y media" –  Gonzalo Medina Dec 5 '11 at 19:44
    
@GonzaloMedina Done, thank you. :) –  Alenanno Dec 5 '11 at 21:36
    
This was enlightening... –  Joze Dec 6 '11 at 9:07
    
As the RAE linked article states, the preposition "de" in the second construction is optional: "Deben ser las ocho y media" is acceptable. I would even say that, at least in my environment (Argentina), it's sliglhtly more common to ommit it. –  leonbloy Dec 6 '11 at 13:24
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