Indexed families of sets

  1. Let TeX Embedding failed! be an indexed family of subsets of a set TeX Embedding failed!. Prove that
    1. TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed!, which means TeX Embedding failed!, therefore TeX Embedding failed! and thus TeX Embedding failed! meaning

      TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed! and hence TeX Embedding failed! therefore TeX Embedding failed! and in fact TeX Embedding failed! meaning

      TeX Embedding failed!

      so

      TeX Embedding failed!
    2. TeX Embedding failed!

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

      TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed! therefore TeX Embedding failed! which means TeX Embedding failed!, and thus TeX Embedding failed! and

      TeX Embedding failed!

      so it follows that

      TeX Embedding failed!
  2. Let TeX Embedding failed!, TeX Embedding failed! be two indexed families of subsets of a set TeX Embedding failed!. Prove the following:
    1. For each TeX Embedding failed!, TeX Embedding failed!

      Clearly if TeX Embedding failed! then TeX Embedding failed! because TeX Embedding failed!.

    2. For each TeX Embedding failed!, TeX Embedding failed!

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

    3. TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed!. If TeX Embedding failed! then TeX Embedding failed!, and if TeX Embedding failed! then TeX Embedding failed! therefore TeX Embedding failed!. Therefore

      TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed! or TeX Embedding failed! therefore TeX Embedding failed! or TeX Embedding failed!, in either case the implication is that TeX Embedding failed! and it follows that TeX Embedding failed! therefore

      TeX Embedding failed!

      so that

      TeX Embedding failed!
    4. TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed! but this means that TeX Embedding failed! and TeX Embedding failed! for any TeX Embedding failed! and hence TeX Embedding failed! and TeX Embedding failed!, therefore TeX Embedding failed! and

      TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed! but this means that TeX Embedding failed! and TeX Embedding failed!, therefore for any TeX Embedding failed!, TeX Embedding failed! and TeX Embedding failed! so that TeX Embedding failed! and thus TeX Embedding failed!, hence TeX Embedding failed! so that

      TeX Embedding failed!

      it follows then that

      TeX Embedding failed!
  3. If for each TeX Embedding failed!, TeX Embedding failed! then
    1. TeX Embedding failed!,

      If TeX Embedding failed! then TeX Embedding failed! and since for each TeX Embedding failed!, TeX Embedding failed! it follows that TeX Embedding failed! and in fact TeX Embedding failed!.

    2. TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed! and since TeX Embedding failed! this means TeX Embedding failed! and thus TeX Embedding failed!.

  4. Let TeX Embedding failed!. Then
    1. TeX Embedding failed!

      Direct proof:

      Let TeX Embedding failed!, then TeX Embedding failed!, now since TeX Embedding failed! TeX Embedding failed! and

      TeX Embedding failed!

      Let TeX Embedding failed!, then TeX Embedding failed! and TeX Embedding failed!. From the latter it follows that TeX Embedding failed! and thus, given TeX Embedding failed!, TeX Embedding failed! from which it follows that TeX Embedding failed! and

      TeX Embedding failed!

      leading to the conclusion

      TeX Embedding failed!

      Alternative proof, proof by contradiction:

      Let TeX Embedding failed!, it follows that TeX Embedding failed! and TeX Embedding failed!. Assume that TeX Embedding failed!, then either TeX Embedding failed! or TeX Embedding failed!. TeX Embedding failed! so the former is impossible, thus the latter must be true which means that TeX Embedding failed! but this is a contradiction. Therefore the assumption is false and in fact TeX Embedding failed!.

      Let TeX Embedding failed!. Then TeX Embedding failed! and TeX Embedding failed!. Assume that TeX Embedding failed! then TeX Embedding failed!. Since TeX Embedding failed!, it follows that TeX Embedding failed!, which is a constradiction. Therefore the assumption is false and in fact TeX Embedding failed!

    2. TeX Embedding failed!

      Let TeX Embedding failed!. Assume TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed!, the latter means TeX Embedding failed!, and it follows from TeX Embedding failed! that TeX Embedding failed!, therefore TeX Embedding failed! but this is a contradiction. Therefore it is not true that TeX Embedding failed! and it follows that

      TeX Embedding failed!

      Let TeX Embedding failed!. Assume TeX Embedding failed!, then TeX Embedding failed!, considering this TeX Embedding failed!, TeX Embedding failed! which means that TeX Embedding failed! and TeX Embedding failed!. Since TeX Embedding failed!, it follows from TeX Embedding failed! that TeX Embedding failed! but TeX Embedding failed! so this is a contradiction. Therefore the assumption is false and in fact TeX Embedding failed! so that

      TeX Embedding failed!

      leading to the conclusion that

      TeX Embedding failed!
  5. Let TeX Embedding failed!, TeX Embedding failed!, TeX Embedding failed!. Then
    1. TeX Embedding failed!,

      If TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed!, therefore TeX Embedding failed! and TeX Embedding failed! or TeX Embedding failed!, if TeX Embedding failed! then TeX Embedding failed! hence TeX Embedding failed!, and if TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed! so in either case TeX Embedding failed! so

      TeX Embedding failed!

      If TeX Embedding failed! then either TeX Embedding failed! or TeX Embedding failed!. If TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed!, from the latter it follows that TeX Embedding failed! and hence TeX Embedding failed!. And if TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed!, and from the latter TeX Embedding failed! and thus TeX Embedding failed!. So

      TeX Embedding failed!

      and

      TeX Embedding failed!
    2. TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed! or TeX Embedding failed!. If TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed! so TeX Embedding failed!, if TeX Embedding failed! then TeX Embedding failed! so TeX Embedding failed! and TeX Embedding failed! so TeX Embedding failed!, therefore TeX Embedding failed! and

      TeX Embedding failed!

      If TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed!, it is clear that if TeX Embedding failed! then TeX Embedding failed! and TeX Embedding failed! so that TeX Embedding failed! satisfies the relation. If TeX Embedding failed! then TeX Embedding failed! must be in TeX Embedding failed! in the first clause and TeX Embedding failed! must be in TeX Embedding failed! in the second clause so that TeX Embedding failed! is in both clauses. Therefore either TeX Embedding failed! or TeX Embedding failed! hence TeX Embedding failed! and

      Let TeX Embedding failed! and assume TeX Embedding failed!. It follows from the assumption that TeX Embedding failed! and TeX Embedding failed!. TeX Embedding failed!, yet from the premise TeX Embedding failed!, therefore TeX Embedding failed!, similarly TeX Embedding failed! and hence TeX Embedding failed!. Therefore TeX Embedding failed! and TeX Embedding failed! and thus TeX Embedding failed! which is a contradiction. Therefore the assumption is false and in fact TeX Embedding failed! so

      TeX Embedding failed!

      and

      TeX Embedding failed!
  6. Let TeX Embedding failed! be an indexed family of subsets of a set TeX Embedding failed!. Let TeX Embedding failed!. Prove that
    1. TeX Embedding failed!.

      If TeX Embedding failed! then TeX Embedding failed! and since TeX Embedding failed!, TeX Embedding failed!, hence TeX Embedding failed!. Therefore TeX Embedding failed!, or in other words TeX Embedding failed!.

    2. TeX Embedding failed!.

      Let TeX Embedding failed!, then TeX Embedding failed! but since TeX Embedding failed!, TeX Embedding failed!, hence TeX Embedding failed! and TeX Embedding failed!.

  7. Let TeX Embedding failed! be an indexed family of subsets of a set TeX Embedding failed!. Let TeX Embedding failed!. Prove that
    1. TeX Embedding failed! if and only if for each TeX Embedding failed!, TeX Embedding failed!.

      Let TeX Embedding failed! and assume TeX Embedding failed!. From the assumption it follows that TeX Embedding failed!, consider this TeX Embedding failed!, since TeX Embedding failed!, TeX Embedding failed!, thus considering this TeX Embedding failed! it follows that TeX Embedding failed! and TeX Embedding failed! which means TeX Embedding failed! which contradicts the premise. Therefore the assumption is false and

      TeX Embedding failed!

      Now let TeX Embedding failed! and assume TeX Embedding failed! is not true. From the assumption it follows that TeX Embedding failed!, considering this TeX Embedding failed!, TeX Embedding failed!, and hence TeX Embedding failed! which means TeX Embedding failed! but this contradicts the premise, hence the assumption is false and

      TeX Embedding failed!

      and therefore

      TeX Embedding failed!
    2. TeX Embedding failed! if and only if for each TeX Embedding failed!, TeX Embedding failed!.

      Let TeX Embedding failed!, TeX Embedding failed! and assume TeX Embedding failed!. From the assumption it follows that TeX Embedding failed!. Consider this TeX Embedding failed!, since TeX Embedding failed!, it follows that TeX Embedding failed!, yet TeX Embedding failed! and hence TeX Embedding failed!, but this is a contradiction. Therefore the assumption is false and in fact

      TeX Embedding failed!

      Now let TeX Embedding failed! and assume that TeX Embedding failed! is false. From the assumption it follows that TeX Embedding failed!. Considering this TeX Embedding failed! it follows that TeX Embedding failed!, but this means TeX Embedding failed! and hence TeX Embedding failed! which contradicts the premise. Therefore the assumption is false and in fact

      TeX Embedding failed!

      which means

      TeX Embedding failed!
  8. Let TeX Embedding failed! be the set of real numbers that are greater than 0. For each TeX Embedding failed!, let TeX Embedding failed! be the open interval TeX Embedding failed!. Prove that
    1. TeX Embedding failed!.

      TeX Embedding failed! hence if TeX Embedding failed!, since TeX Embedding failed! and TeX Embedding failed!, it follows that TeX Embedding failed!. Assume that TeX Embedding failed!, then TeX Embedding failed!. Consider this TeX Embedding failed!. Clearly TeX Embedding failed!, yet TeX Embedding failed! which means TeX Embedding failed!, but this is a contradiction. Therefore, the assumption is false and infact TeX Embedding failed!.

    2. TeX Embedding failed!.

      Let TeX Embedding failed!, then TeX Embedding failed! and hence TeX Embedding failed! since TeX Embedding failed!, thus TeX Embedding failed!.

      Let TeX Embedding failed!, clearly TeX Embedding failed! and since TeX Embedding failed! it follows that TeX Embedding failed! and so TeX Embedding failed!.

      Therefore TeX Embedding failed!.

  9. Let TeX Embedding failed! be the set of real numbers that are greater than TeX Embedding failed!. For each TeX Embedding failed!, let TeX Embedding failed! be the closed interval TeX Embedding failed!. Prove that
    1. TeX Embedding failed!.

      TeX Embedding failed! because TeX Embedding failed!. Thus, since TeX Embedding failed! is the only element of TeX Embedding failed!, TeX Embedding failed! and thus

      TeX Embedding failed!

      I say that if TeX Embedding failed!, then TeX Embedding failed!. Proof, assume that TeX Embedding failed! and consider this TeX Embedding failed!. TeX Embedding failed! yet TeX Embedding failed! and hence TeX Embedding failed!, but this is a contradiction. Therefore, the assumption is false and in fact TeX Embedding failed! and hence if TeX Embedding failed! then TeX Embedding failed!, consequently TeX Embedding failed! and so

      TeX Embedding failed!

      which leads to the conclusion

      TeX Embedding failed!
    2. TeX Embedding failed!.

      Let TeX Embedding failed! then TeX Embedding failed! and hence TeX Embedding failed! for some TeX Embedding failed!. TeX Embedding failed! is therefore either TeX Embedding failed!, in which case TeX Embedding failed! or TeX Embedding failed! is a real number which is greater than TeX Embedding failed! and less than or equal to TeX Embedding failed!, in which case TeX Embedding failed! and thus TeX Embedding failed!, therefore

      TeX Embedding failed!

      Let TeX Embedding failed! then either TeX Embedding failed!, in which case TeX Embedding failed! (TeX Embedding failed! is an element of any TeX Embedding failed!, where TeX Embedding failed!), or TeX Embedding failed! in which case TeX Embedding failed! and thus TeX Embedding failed!, therefore

      TeX Embedding failed!

      so that

      TeX Embedding failed!