Skip to content

DA1. Operations on Sets

Statement

  • Create three sets A, B having 4 elements in each, and U, a Universal set of any possible number of elements of your interest.
  • Then explain the answers to the following questions to your peers:
    1. \(A ∪ B\)
    2. \(A ∩ B\)
    3. \((A ∩ B) ∪ U\)
    4. The power set of A.
    5. \(A'\).
    6. \(∅ ∩ B\).
    7. \(A × B\).
    8. \(A - B\).
    9. \((A - B) ∪ (B - A)\).
    10. Prove any one De Morgan identity for A and B.

Solution

We will use the following sets:

  • \(A = \{1, 2, 3, 4\}\)
  • \(B = \{3, 5, 6\}\)
  • \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)

Answer 1

  • \(A ∪ B = \{1, 2, 3, 4, 5, 6\}\)
  • Reads as “A union B”, which is the set of all elements that are in A or B or both (either A or B).

Answer 2

  • \(A ∩ B = \{3\}\)
  • Reads as “A intersect B”, which is the set of all elements that are in both A and B (both A and B).

Answer 3

\[ \begin{aligned} (A ∩ B) ∪ U &= \{3\} ∪ U \\ &= U \\ &= \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \end{aligned} \]
  • Reads as “(A intersect B) union U”, which is the set of all elements that are in both A and B or in U.

Answer 4

\[ \begin{aligned} \mathcal{p}(A) = \{ \\ &\,\, \emptyset, \\ &\,\, \{1\}, \{2\}, \{3\}, \{4\}, \\ &\,\, \{1, 2\}, \{1, 3\}, \{1, 4\}, \{2, 3\}, \{2, 4\}, \{3, 4\}, \\ &\,\, \{1, 2, 3\}, \{1, 2, 4\}, \{1, 3, 4\}, \{2, 3, 4\}, \\ &\,\, \{1, 2, 3, 4\} \\ \} \end{aligned} \]
  • The power set of A is the set of all possible subsets of A.
  • The empty set is a subset of every set, so it is included in \(\mathcal{p}(A)\).
  • The subsets that have the same elements but are in different order do not count as different subsets, so \(\{1, 2\}\) and \(\{2, 1\}\) are the same subset.

Answer 5

  • \(A' = \{5, 6, 7, 8, 9, 10\}\)
  • Reads as “the complement of A”, and can be represented as \(\bar{A}\) or \(A^c\), and includes all elements that are not in A, but do exist in the universal set U.
  • In other words, \(A' = U - A = \{5, 6, 7, 8, 9, 10\}\).

Answer 6

  • \(∅ ∩ B = ∅\)
  • Reads as “the empty set intersect B”, which is the set of all elements that are in both the empty set and B.
  • This statement always returns the empty set, no matter what B is.

Answer 7

\[ \begin{aligned} A × B &= \{ \\ &\,\, (1, 3), (1, 5), (1, 6), \\ &\,\, (2, 3), (2, 5), (2, 6), \\ &\,\, (3, 3), (3, 5), (3, 6), \\ &\,\, (4, 3), (4, 5), (4, 6) \\ \} \end{aligned} \]
  • Reads as “A times B” which results in the cartesian product of A and B, which in turn means all possible ordered pairs of elements from A and B.

Answer 8

  • \(A - B = \{1, 2, 4\}\)
  • Reads as “A minus B”, which is the set of all elements that are in A but not in B.

Answer 9

\[ \begin{aligned} (A - B) ∪ (B - A) &= \{1, 2, 4\} ∪ \{5, 6\} \\ &= \{1, 2, 4, 5, 6\} \end{aligned} \]
  • Reads as “(A minus B) union (B minus A)”, which is the set of all elements that are in A but not in B or in B but not in A.

Answer 10

  • De Morgan’s Law Includes two dualities:
    1. \((A ∪ B)' = A' ∩ B'\)
    2. \((A ∩ B)' = A' ∪ B'\)
  • I will use the Venn diagram below to prove the identity.
  • (1). We can compare the colored area representing \((A ∪ B)'\) with the colored area representing \(A' ∩ B'\), and find that they are the same.
  • (2). We can also compare the colored area representing \((A ∩ B)'\) with the colored area representing \(A' ∪ B'\), and find that they are the same.

References

  • Doerr, A., & Levasseur, K. (2022). Applied discrete structures (3rd ed.). licensed under CC BY-NC-SA. Chapter 4: More on Sets.
  • Levin, O. (2021). Discrete mathematics: An open introduction (3rd ed.). licensed under CC 4.0. Chapter 0: Introduction.