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Math Help - Set theory doubt: intersection and union

  1. #1
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    Set theory doubt: intersection and union

    We can define Union, within sets A within a set P, by the expression:

    {x| x E A for some A E P}

    The expression for intersection goes as:

    {x |x E A for all A E P}

    Why does union use the operator "for some" and intersection use the operator "for all"?
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  2. #2
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    Re: Set theory doubt: intersection and union

    Because union of sets contains elements that belong to some of those sets, and intersection contains elements that belong to all sets. This is the definition.
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  3. #3
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    Re: Set theory doubt: intersection and union

    Suppose we know an element $x$ is in a union of sets. We have no way of knowing for this, "which ones" it may lie within. At least one, but maybe more. But AT LEAST one. In this instance, "...for some..." means "there exists....".

    Alternatively, suppose we know $x$ lies in an intersection of sets. Then it lies in every single set we are taking the intersection of.

    In other words, "there exists" is a sort of generalized (universal) "or", and "for all" is a sort of generalized "and".
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