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!
    2. 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!
    2. For each TeX Embedding failed!, TeX Embedding failed!
    3. TeX Embedding failed!
    4. TeX Embedding failed!
  3. If for each TeX Embedding failed!, TeX Embedding failed! then
    1. TeX Embedding failed!,
    2. TeX Embedding failed!
  4. Left TeX Embedding failed!. Then
    1. TeX Embedding failed!,
    2. TeX Embedding failed!
  5. Let TeX Embedding failed!, TeX Embedding failed!, TeX Embedding failed!. Then
    1. TeX Embedding failed!,
    2. 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!.
    2. 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!.
    2. TeX Embedding failed! if and only if for each TeX Embedding failed!, 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!.
    2. TeX Embedding failed!.
  9. For each TeX Embedding failed!, let TeX Embedding failed! be the closed interval TeX Embedding failed!. Prove that
    1. TeX Embedding failed!.
    2. TeX Embedding failed!.