For all sets A, B and C Is (A B) C=A (B C) ? Justify your stateme… - Vidyadip

Question

For all sets A, B and C
Is $(A \cap B) \cup C=A \cap(B \cup C)$ ?
Justify your statement.

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Answer

Let us consider the following sets A, B and C such that
$\begin{array}{l} A =\{1,2,3\} \\ B =\{2,3,5\} \\ C =\{4,5,6\}\end{array}$
$\begin{array}{l}\text { Now }(A \cap B) \cup C=(\{1,2,3\} \cap\{2,3,5\}) \cup\{4,5,6\} \\ =\{2,3\} \cup\{4,5,6\} \\ =\{2,3,4,5,6\}\end{array}$
$\begin{array}{l}\text { And } A \cap(B \cup C)=\{1,2,3\} \cap[\{2,3,5\} \cup\{4,5,6\} \\ =\{1,2,3\} \cap\{2,3,4,5,6\}\end{array}$
$\begin{array}{l}=\{2,3\} \\ \text { Thus, }(A \cap B) \cup C \neq A \cap(B \cup C)\end{array}$

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