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cs-401r:set-theory-identities [2014/09/09 14:44]
ringger
cs-401r:set-theory-identities [2014/09/09 21:00]
ringger [Other Useful Identities]
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 += Set Theory Identities =
 \begin{align} \begin{align}
 A \cup  \overline{A} & = & \Omega ​ \qquad & \mbox{Complementation law}\\ A \cup  \overline{A} & = & \Omega ​ \qquad & \mbox{Complementation law}\\
 A \cap  \overline{A} & = & \emptyset ​  ​\qquad & \mbox{Exclusion law}\\ \\ A \cap  \overline{A} & = & \emptyset ​  ​\qquad & \mbox{Exclusion law}\\ \\
-A \cap \Omega &\equiv ​& A \qquad & \mbox{Identity laws} \\ +A \cap \Omega & & A \qquad & \mbox{Identity laws} \\ 
-A \cup \emptyset& ​\equiv ​& A \qquad & \\ \\ +A \cup \emptyset& ​& A \qquad & \\ \\ 
-A \cup \Omega &\equiv ​& \Omega \qquad & \mbox{Domination laws} \\ +A \cup \Omega && \Omega \qquad & \mbox{Domination laws} \\ 
-A \cap \emptyset &\equiv ​& \emptyset \qquad &  \\ \\ +A \cap \emptyset & & \emptyset \qquad &  \\ \\ 
-A \cup A &\equiv ​& A \qquad & \mbox{Idempotent laws} \\ +A \cup A & & A \qquad & \mbox{Idempotent laws} \\ 
-A \cap A &\equiv ​& A \qquad &  \\ \\ +A \cap A & & A \qquad &  \\ \\ 
-\overline{\left(\overline{A}\right)} &\equiv ​& A \qquad & \mbox{Double Complement} \\ \\ +\overline{\left(\overline{A}\right)} & & A \qquad & \mbox{Double Complement} \\ \\ 
-A \cup B &\equiv ​& B \cup A \qquad & \mbox{Commutative laws} \\ +A \cup B & & B \cup A \qquad & \mbox{Commutative laws} \\ 
-A \cap B &\equiv ​& B \cap A \qquad &  \\ \\ +A \cap B & & B \cap A \qquad &  \\ \\ 
-\left(A \cup B\right) \cup C &\equiv ​& A \cup \left(B \cup C\right) \qquad & \mbox{Associative laws} \\ +\left(A \cup B\right) \cup C & & A \cup \left(B \cup C\right) \qquad & \mbox{Associative laws} \\ 
-\left(A \cap B\right) \cap C &\equiv ​& A \cap \left(B \cap C\right) \qquad & \\ \\ +\left(A \cap B\right) \cap C & & A \cap \left(B \cap C\right) \qquad & \\ \\ 
-A \cup \left(B \cap C\right) &\equiv ​& \left(A \cup B\right) \cap \left(A \cup C\right) \qquad & \mbox{Distributive laws} \\ +A \cup \left(B \cap C\right) & & \left(A \cup B\right) \cap \left(A \cup C\right) \qquad & \mbox{Distributive laws} \\ 
-A \cap \left(B \cup C\right) &\equiv ​& \left(A \cap B\right) \cup \left(A \cap C\right) \qquad \\ \\ +A \cap \left(B \cup C\right) & & \left(A \cap B\right) \cup \left(A \cap C\right) \qquad \\ \\ 
-\overline{\left(A \cap B \right)} &\equiv ​& \overline{A} \cup \overline{B} \qquad & \mbox{De Morgans laws} \\ +\overline{\left(A \cap B \right)} & & \overline{A} \cup \overline{B} \qquad & \mbox{De Morgans laws} \\ 
-\overline{\left(A \cup B \right)} &\equiv ​& \overline{A} \cap \overline{B} \qquad & \\+\overline{\left(A \cup B \right)} & & \overline{A} \cap \overline{B} \qquad & \\
  
 \end{align} \end{align}
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 \begin{align} \begin{align}
-B - A \equiv B \cap \overline{A} \\ +B - A \equiv B \cap \overline{A} \qquad & \mbox{Definition of set difference} \\ 
-\left(R \cap S\right) \cup \left(R \cap \overline{S}\right) ​\equiv ​R+\left(R \cap S\right) \cup \left(R \cap \overline{S}\right) ​R
 \end{align} \end{align}
  
cs-401r/set-theory-identities.txt · Last modified: 2014/10/06 07:08 by ringger
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