Differences

This shows you the differences between two versions of the page.

 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] Both sides previous revision Previous revision 2014/10/06 07:08 ringger [Further Reference] 2014/09/09 21:00 ringger [Other Useful Identities] 2014/09/09 20:57 ringger 2014/09/09 14:52 ringger 2014/09/09 14:44 ringger 2014/09/09 14:44 ringger 2014/09/05 08:09 ringger 2014/09/05 08:09 ringger created Next revision Previous revision 2014/10/06 07:08 ringger [Further Reference] 2014/09/09 21:00 ringger [Other Useful Identities] 2014/09/09 20:57 ringger 2014/09/09 14:52 ringger 2014/09/09 14:44 ringger 2014/09/09 14:44 ringger 2014/09/05 08:09 ringger 2014/09/05 08:09 ringger created Last revision Both sides next revision Line 1: Line 1: + = Set Theory Identities = \begin{align} \begin{align} - A \cup  \overline{A} & \equiv ​& \Omega  ​= & \mbox{Complementation law}\\ + A \cup  \overline{A} & = & \Omega  ​\qquad ​& \mbox{Complementation law}\\ - A \cap  \overline{A} & \equiv ​& \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} Line 23: Line 24: \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
Back to top