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Mathematics LibreTexts

2.3: Properties of Sets

  • Page ID
    4873
  • [ "article:topic", "authorname:thangarajahp" ]

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     Let \(A, B,\) and \(C\) be sets and \(U\) be the universal set. Then:

    Commutativity

    1. \(A \cup B =B \cup A \) and

        \(A \cap B= B \cap A \)

    Proof: 

    Let \(x \in A \cup B\). Then \(x \in A\) or \(x \in B\). Which implies \(x \in B\) or \(x \in A\). Hence \(x \in B \cup A\). Thus \(A \cup B \subseteq B \cup A \). Similarly, we can show that \(B \cup A \subseteq A \cup B \). Therefore, \(A \cup B =B \cup A \).

     

    Let \(x \in A \cap B\). Then \(x \in A\) and \(x \in B\). Which implies \(x \in B\) and \(x \in A\). Hence \(x \in B \cap A\). Thus \(A \cap B \subseteq B \cap A \). Similarly, we can show that \(B \cap A \subseteq A \cap B \). Therefore, \(A \cap B =B \cap A \).

    Distributivity

    2. \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) and \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C).\)

    We have illustrated using a Venn diagram:

    Example \(\PageIndex{1}\):

    Consider \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\):

    De Morgan's Laws

    3. \((A \cup B)^c = A^c \cap B^c\) and \((A \cap B)^c = A^c \cup B^c \)

    Example \(\PageIndex{2}\):

    Consider \((A \cup B)^c = A^c \cap B^c\):

    Relative Complements

    4. \(A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)\) and \(A \setminus (B \cap C) = (A \setminus B) \cup (A \setminus C).\)

    Example \(\PageIndex{3}\):

    Consider \(A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)\):

    Others 

    5. \(A \cap A=A\) and \(A \cup A=A\).

    6. \(A \cap \emptyset= \emptyset\) and \(A \cup \emptyset=A\).

    7. \(A \cap A^c= \emptyset\) and \(A \cup A^c= U\).

    8. \((A^c)^C=A\).

    9. \(\emptyset^c=U\).

    10. \(U^c=\emptyset\).