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Mathematics LibreTexts

2.3: Properties of Sets

  • Page ID
    4873
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    Let \(A, B,\) and \(C\) be sets and \(U\) be the universal set. Then:

    Commutative Law

    1. \(A \cup B =B \cup A \) and

    \(A \cap B= B \cap A \)

    Theorem \(\PageIndex{1}\): Commutative Law

    For all sets \(A\) and \(B\), \(A \cup B =B \cup A \) 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, \(A \cap B =B \cap A \).

    Distributive Law

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

    Theorem \(\PageIndex{2}\): Distributive Law

    For all sets \(A,B \) and \(C\), \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) and \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C).\)

    Proof

    Let \( x \in \(A \cap (B \cup C) \).

    Then  \(x \in A\) and  \( x \in B \cup C\).

    Thus  \(x \in A\) and  \( x \in B \) or \(x \in C\).

    Which implies \(x \in A\) and  \( x \in B \) or  \(x \in A\) and  \( x \in C \).

    Hence  \( x \in (A \cap B) \cup (A \cap C)\).  Thus  \(A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)\).  Similarly, we can show that \((A \cap B) \cup (A \cap C) \subseteq A \cap (B \cup C) \). Therefore, \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\).

    De Morgan's Laws

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

    We have illustrated using a Venn diagram:

    Example \(\PageIndex{2}\):

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

    Theorem \(\PageIndex{3}\)

    For all sets \(A\) and \(B\),  \((A \cup B)^c = A^c \cap B^c\) and \((A \cap B)^c = A^c \cup B^c \).

    Proof

    Exercise.

     

    Relative Complements

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

    We have illustrated using a Venn diagram:

    Example \(\PageIndex{3}\):

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

    Proof:

    Discussed in class.

    Idempotents


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

    Identity

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

    Complements

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

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

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

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