Exercise 1: (i) Y∅ has exactly one element, namely ∅, whether Y is empty or not, and (ii) if X is not empty, then ∅ˣ is empty.

Exercise 2: Prove that (∪ᵢ Aᵢ) × (∪ⱼ Bⱼ) = ∪ᵢ,ⱼ (Aᵢ × Bⱼ) and that the same equation holds for intersections (provided that the domains of the families involved are not empty). Prove also (with appropriate provisos about empty families) that ∩ᵢ Xᵢ ⊂ Xⱼ ⊂ ∪ᵢ Xᵢ for each index j and that intersection and union can in fact be characterized as the extreme solutions of these inclusions. This means that if Xⱼ ⊂ Y for each index j, then ∪ᵢ Xᵢ ⊂ Y, and that ∪ᵢ Xᵢ is the only set satisfying the minimality condition; the formulation for intersections is similar.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    50% discount

    $8.00 $4.00
    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Advanced Math Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats