Set operations: Union, Intersection, and Compliment

  1. Prove that TeX Embedding failed!

    If TeX Embedding failed! then TeX Embedding failed! so TeX Embedding failed! and TeX Embedding failed!, which means that TeX Embedding failed! and TeX Embedding failed! so that TeX Embedding failed!, therefore

    TeX Embedding failed!

    If TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed! so TeX Embedding failed!, hence TeX Embedding failed!, therefore

    TeX Embedding failed!

    it follows that

    TeX Embedding failed!
  2. Prove that TeX Embedding failed!

    It is easy to prove this from the previous proof (as noted by Bert himself), namely that TeX Embedding failed!, taking the complements of TeX Embedding failed! and TeX Embedding failed!

    TeX Embedding failed!

    taking complements of the first and last terms

    TeX Embedding failed!

    so that

    TeX Embedding failed!

    Alternatively, it can be proved from scratch:

    If TeX Embedding failed! then TeX Embedding failed!, so at least one of TeX Embedding failed! or TeX Embedding failed! must be true, thus TeX Embedding failed! and

    TeX Embedding failed!

    If TeX Embedding failed! then at least one of TeX Embedding failed! or TeX Embedding failed! must be true, therefore it is never true that TeX Embedding failed! and hence it is always true that TeX Embedding failed! and

    TeX Embedding failed!

    so that

    TeX Embedding failed!

    Using the same technique as before, this relation can be used to prove that TeX Embedding failed!. Taking the complements of TeX Embedding failed! and TeX Embedding failed! in TeX Embedding failed! gives

    TeX Embedding failed!

    and taking complements of the first and last terms gives

    TeX Embedding failed!

    and it follows that TeX Embedding failed!.

  3. Let TeX Embedding failed!, TeX Embedding failed!. Prove the following:
    1. TeX Embedding failed! if and only if TeX Embedding failed!.

      First, the if part. Let TeX Embedding failed! and assume TeX Embedding failed!. From the assumption it follows that TeX Embedding failed! and consequently TeX Embedding failed!. This means TeX Embedding failed! and it follows that TeX Embedding failed! which is a contradiction. Thus the assumption is false and in fact TeX Embedding failed!.

      Second, the only if part. Let TeX Embedding failed! and assume TeX Embedding failed!. From the assumption, since TeX Embedding failed!, to prevent equality it follows that TeX Embedding failed! which means TeX Embedding failed!. Consider such an TeX Embedding failed!, TeX Embedding failed! but TeX Embedding failed!, therefore TeX Embedding failed! and TeX Embedding failed! which means TeX Embedding failed! which is a contradiction. Thus the assumption is false and in fact TeX Embedding failed!.

    2. TeX Embedding failed! if and only if TeX Embedding failed!.

      Let TeX Embedding failed! and assume TeX Embedding failed!. From the assumption it follows that TeX Embedding failed!, but this means that TeX Embedding failed! which means TeX Embedding failed! which is a contradiction. Hence the assumption is false and in fact TeX Embedding failed!.

      If TeX Embedding failed!, then TeX Embedding failed!. Let TeX Embedding failed! and assume that TeX Embedding failed!. From the assumption it follows, since TeX Embedding failed!, that TeX Embedding failed!. Thus TeX Embedding failed!, and since TeX Embedding failed!, TeX Embedding failed!. Hence TeX Embedding failed! which is a contradiction. Therefore the assumption is false and in fact TeX Embedding failed!.

    3. TeX Embedding failed! if and only if TeX Embedding failed!.

      If TeX Embedding failed! then TeX Embedding failed! therefore TeX Embedding failed! and TeX Embedding failed!.

      If TeX Embedding failed! then TeX Embedding failed! and therefore TeX Embedding failed!.

    4. TeX Embedding failed! if and only if TeX Embedding failed! (TeX Embedding failed! is the universe).

      If TeX Embedding failed! then TeX Embedding failed! and therefore TeX Embedding failed!.

      If TeX Embedding failed! then TeX Embedding failed! so for any TeX Embedding failed! either TeX Embedding failed! or TeX Embedding failed!, or in other words TeX Embedding failed!.

      Alternative proof:

      Let TeX Embedding failed!, and assume TeX Embedding failed!, then TeX Embedding failed!, and so TeX Embedding failed!.

      Let TeX Embedding failed!, and assume TeX Embedding failed!. TeX Embedding failed! is the universe so TeX Embedding failed! and thus it follows from the assumption that TeX Embedding failed!, therefore TeX Embedding failed!, taking this TeX Embedding failed!, since TeX Embedding failed!, by TeX Embedding failed!, it follows that TeX Embedding failed!, but this is a contradiction since TeX Embedding failed!. Therefore the assumption is false and in fact TeX Embedding failed!.

    5. TeX Embedding failed! if and only if TeX Embedding failed!.

      Let TeX Embedding failed! and assume that TeX Embedding failed! then TeX Embedding failed!, but this means that TeX Embedding failed!, or in other words TeX Embedding failed! and thus TeX Embedding failed! which is a contradiction, therefore the assumption is false and in fact TeX Embedding failed!.

      Let TeX Embedding failed! and assume TeX Embedding failed!. From the assumption, TeX Embedding failed! and thus TeX Embedding failed!, or in other words TeX Embedding failed! and hence TeX Embedding failed! which is a contradiction. Hence the assumption is false and in fact TeX Embedding failed!.

      Alternative proof:

      Let TeX Embedding failed!. Assume TeX Embedding failed!, then TeX Embedding failed!, consider this TeX Embedding failed!, since TeX Embedding failed!, by TeX Embedding failed!, TeX Embedding failed!, which is a contradiction. Therefore, the assumption is false and in fact TeX Embedding failed!.

      Let TeX Embedding failed!. Assume TeX Embedding failed! then TeX Embedding failed!, consider this TeX Embedding failed!, since TeX Embedding failed!, by TeX Embedding failed!, TeX Embedding failed!, which is a contradiction. Therefore, the assumption is false and in fact TeX Embedding failed!.

    6. TeX Embedding failed! if and only if TeX Embedding failed!.

      Let TeX Embedding failed!. Assume TeX Embedding failed!. From the assumption TeX Embedding failed! or in other words TeX Embedding failed! which in other words says TeX Embedding failed! and hence TeX Embedding failed! which is a contradiction. Therefore the assumption is false and in fact TeX Embedding failed!.

      Let TeX Embedding failed!. Assume TeX Embedding failed!. From the assumption TeX Embedding failed! or in other words TeX Embedding failed! which in other words says TeX Embedding failed! and hence TeX Embedding failed! which is a contradiction. Therefore the assumption is false and in fact TeX Embedding failed!.

      Alternative proof:

      Let TeX Embedding failed!. Assume TeX Embedding failed!, then TeX Embedding failed!. Consider this TeX Embedding failed!, since TeX Embedding failed!, TeX Embedding failed! which is a contradiction. Therefore the assumption is false and in fact TeX Embedding failed!.

      Let TeX Embedding failed!. Assume TeX Embedding failed!, then TeX Embedding failed!. Consider this TeX Embedding failed!, since TeX Embedding failed!, TeX Embedding failed!, which is a contradiction. Therefore the assumption is false and in fact TeX Embedding failed!.

  4. Let TeX Embedding failed!. Prove the following:
    1. TeX Embedding failed!.

      If TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed!, but since TeX Embedding failed! then TeX Embedding failed!, and given TeX Embedding failed! it follows that TeX Embedding failed! and hence TeX Embedding failed!.

    2. TeX Embedding failed!

      I was humbled when a former colleague Michael Pfeiffer showed me a trivial proof for this, after I had written a one page proof. So here it is:

      Let TeX Embedding failed! and let TeX Embedding failed!

      TeX Embedding failed!, so TeX Embedding failed!, and

      TeX Embedding failed!

      and (the first step is by Demorgan's laws):

      TeX Embedding failed!

      since TeX Embedding failed! and TeX Embedding failed!, it follows that

      TeX Embedding failed!

      Now here is the drawn out proof I began with:

      Preliminaries: generally TeX Embedding failed! and thus TeX Embedding failed!. Generally TeX Embedding failed! and thus TeX Embedding failed! which means that generally

      \begin{equation} \label{distributivesetsubtraction} (A \cup B) - C = (A - C) \cup (B - C) \end{equation}

      It is required to prove that TeX Embedding failed!.

      Let TeX Embedding failed!, then TeX Embedding failed! and TeX Embedding failed!. Another way to write TeX Embedding failed! is to write TeX Embedding failed! since TeX Embedding failed! (when TeX Embedding failed! as is the case here). Therefore TeX Embedding failed! and TeX Embedding failed!, which is alternatively written TeX Embedding failed!. Now, since generally TeX Embedding failed!, it follows that TeX Embedding failed!. Now since TeX Embedding failed! it follows that TeX Embedding failed!, and since TeX Embedding failed!, TeX Embedding failed!, hence TeX Embedding failed! and hence

      TeX Embedding failed!

      Now let TeX Embedding failed!, then either TeX Embedding failed! or TeX Embedding failed!.

      First case, let TeX Embedding failed!, then since TeX Embedding failed!, TeX Embedding failed! and therefore TeX Embedding failed!. Now assume TeX Embedding failed!, since TeX Embedding failed! for TeX Embedding failed! to be true it must follow that TeX Embedding failed!, but this leads to a contradiction because if TeX Embedding failed!, then TeX Embedding failed! but it was stated that TeX Embedding failed!. So the assumption is false and in fact TeX Embedding failed! and

      TeX Embedding failed!

      Second case, let TeX Embedding failed!. It is true that TeX Embedding failed! since TeX Embedding failed!. Assume that TeX Embedding failed!, then since TeX Embedding failed! it must be true that TeX Embedding failed!, but TeX Embedding failed! and so TeX Embedding failed!, but this is a contradiction since TeX Embedding failed! means that TeX Embedding failed!. Hence the assumption is false and in fact TeX Embedding failed!. Thus

      TeX Embedding failed!

      so putting the cases together, if TeX Embedding failed!, then TeX Embedding failed!, and if TeX Embedding failed! then TeX Embedding failed!, it follows that if TeX Embedding failed! then TeX Embedding failed! and hence

      TeX Embedding failed!

      and so

      TeX Embedding failed!