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2.5: 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

    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

    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, \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\).

    We have illustrated using a Venn diagram:

    alt

    De Morgan's Laws

    Theorem \(\PageIndex{3}\): De Morgan's Law 

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

    We have illustrated using a Venn diagram:

    alt

     

    Relative Complements

    Theorem \(\PageIndex{4}\): Relative Complements

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

    We have illustrated using a Venn diagram:

    alt

    Idempotents

    Theorem \(\PageIndex{5}\): Idempotents

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

     

    Identity

    Theorem \(\PageIndex{6}\): Identity

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

    Complements

    Theorem \(\PageIndex{7}\): Complements
    1. \(A \cap A^c= \emptyset\) and\(A \cup A^c= U\).
    2. \((A^c)^c=A\).
    3. \(\emptyset^c=U\).
    4.  \(U^c=\emptyset\).

     


    This page titled 2.5: Properties of Sets is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.