Cho các tập hợp$A,\,\,B$ và $C$. Chứng minh rằng

Cho các tập hợp$A,\,\,B$ và $C$. Chứng minh rằng

a) $\left( A\backslash B \right)\cup \left( B\backslash A \right)=\left( A\cup B \right)\backslash \left( A\cap B \right)$

b) $A\backslash \left( B\cap C \right)=\left( A\backslash B \right)\cup \left( A\backslash C \right)$

c) $A\backslash \left( B\cup C \right)=\left( A\backslash B \right)\cap \left( A\backslash C \right)$
 

Nguyễn Tú

New member
a) Ta có

$\begin{array}{l} \forall x,\,\,x \in \left( {A\backslash B} \right) \cup \left( {B\backslash A} \right) \Leftrightarrow \left[ {\begin{array}{*{20}{c}} {x \in A\backslash B}\\ {x \in B\backslash A} \end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {x \in A}\\ {x \notin B} \end{array}} \right.}\\ {\left\{ {\begin{array}{*{20}{c}} {x \in B}\\ {x \notin A} \end{array}} \right.} \end{array}} \right.\\ \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {x \in A}\\ {x \in B} \end{array}} \right.}\\ {\left[ {\begin{array}{*{20}{c}} {x \notin A}\\ {x \notin B} \end{array}} \right.} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {x \in A \cup B}\\ {x \notin A \cap B} \end{array}} \right. \Leftrightarrow \left( {A \cup B} \right)\backslash \left( {A \cap B} \right) \end{array}$

Suy ra $\left( A\backslash B \right)\cup \left( B\backslash A \right)=\left( A\cup B \right)\backslash \left( A\cap B \right)$.

b)

$\begin{array}{l} \forall x,\,\,x \in A\backslash \left( {B \cap C} \right) \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {x \in A}\\ {x \notin B \cap C} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {x \in A}\\ {\left[ {\begin{array}{*{20}{c}} {x \notin B}\\ {x \notin C} \end{array}} \right.} \end{array}} \right.\\ \Leftrightarrow \left[ {\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {x \in A}\\ {x \notin B} \end{array}} \right.}\\ {\left\{ {\begin{array}{*{20}{c}} {x \in A}\\ {x \notin C} \end{array}} \right.} \end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}} {x \in A\backslash B}\\ {x \in A\backslash C} \end{array}} \right. \Leftrightarrow x \in \left( {A\backslash B} \right) \cup \left( {A\backslash C} \right) \end{array}$

c)

$\begin{array}{l} \forall x,\,\,x \in A\backslash \left( {B \cup C} \right) \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {x \in A}\\ {x \notin B \cup C} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {x \in A}\\ \begin{array}{l} x \notin B\\ x \notin C \end{array} \end{array}} \right.\\ \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {x \in A}\\ {x \notin B} \end{array}} \right.}\\ {\left\{ {\begin{array}{*{20}{c}} {x \in A}\\ {x \notin C} \end{array}} \right.} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {x \in A\backslash B}\\ {x \in A\backslash C} \end{array}} \right. \Leftrightarrow x \in \left( {A\backslash B} \right) \cap \left( {A\backslash C} \right) \end{array}$