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Try It Now

Site: Saylor Academy
Course: CS202: Discrete Structures
Book: Try It Now
Printed by: Guest user
Date: Thursday, April 25, 2024, 12:29 PM

Description

Work these exercises to see how well you understand this material.

Table of contents

Exercises

  1. Let A = {0, 2, 3}, B = {2, 3}, C = {1, 4}, and let the universal set be U = { 0, 1 , 2 , 3 , 4 }. List the elements of:
    1. A × B
    2. B × A
    3. A × B × C
    4. U × ∅
    5. A × Ac
    6. B2
    7. B3
    8. B × \wp(B)

  2. List all two-element sets in \wp(\left \{ a, b, c, d \right \}).

  3. How many singleton (one-element) sets are there in \wp (A) if |A| = n ?

  4. Let A = {+ , −} and B = {00, 01, 10, 11}.
    1. List the elements of A × B
    2. How many elements do A4 and (A × B)3 have?

  5. Let A and B be nonempty sets. When are A × B and B × A equal?

 

Source: Al Doerr and Ken Levasseur, http://faculty.uml.edu/klevasseur/ads-latex/ads.pdf
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Solutions

  1. Answer:
    1. {(0, 2), (0, 3), (2, 2), (2, 3), (3, 2), (3, 3)}
    2. {(2, 0), (2, 2), (2, 3), (3, 0), (3, 2), (3, 3)}
    3. {(0, 2, 1), (0, 2, 4), (0, 3, 1), (0, 3, 4), (2, 2, 1), (2, 2, 4), (2, 3, 1), (2, 3, 4), (3, 2, 1), (3, 2, 4), (3, 3, 1), (3, 3, 4)}
    4. ∅
    5. {(0, 1), (0, 4), (2, 1), (2, 4), (3, 1), (3, 4)}
    6. {(2, 2), (2, 3), (3, 2), (3, 3)}
    7. {(2, 2, 2), (2, 2, 3), (2, 3, 2), (2, 3, 3), (3, 2, 2), (3, 2, 3), (3, 3, 2), (3, 3, 3)}
    8. {(2, ∅), (2, {2}), (2, {3}), (2, {2, 3}), (3, ∅), (3, {2}), (3, {3}), (3, {2, 3})}

  2. Answer: {a, b}, {a, c}, {a, d}, {b, c}, {b, d} and {c, d}

  3. Answer: There are n singleton subsets, one for each element.

  4. Answer:
    1. {+00, +01, +10, +11, −00, −01, −10, −11}
    2. 16 and 512

  5. Answer: They are equal when A = B.