2.5: Properties of Sets
- Page ID
- 4873
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:
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:
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:
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
- \(A \cap A^c= \emptyset\) and\(A \cup A^c= U\).
- \((A^c)^c=A\).
- \(\emptyset^c=U\).
- \(U^c=\emptyset\).