Question

If A, B and C are non-empty sets then the intersection of sets is distributive over union is sets is represented as
$\begin{align}
  & a)A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right) \\
 & b)A\cap \left( B\cap C \right)=\left( A\cap B \right)\cap \left( A\cap C \right) \\
 & c)\left( A\cup B \right)\cup C=\left( A\cap C \right)\cup \left( B\cap C \right) \\
 & d)\left( A\cap B \right)\cup C=\left( A\cap C \right)\cap \left( B\cup C \right) \\
\end{align}$

Answer 1

Hint: Now we know that the union intersection of sets is distributive over. Union is showed as $A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$ . We will take set A, B and C of our choice and check the property.

Complete step by step answer:
Now consider 3 sets A, B and C.
Now we know that according to Algebra of Sets $A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$
Hence we have Distributive property of intersection over union is given by $A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$
Now let us take an example for the same and verify the distributive property.
Let us consider set A, B and C such that A = {1, 2, 3, 4, 5}, B ={3, 4, 5, 6, 7} and
C = {5, 6, 7, 8, 9, 10}
Now consider $A\cap \left( B\cup C \right)$
Let us first find $\left( B\cup C \right)$
Now we have B ={3, 4, 5, 6, 7} and C = {5, 6, 7, 8, 9, 10}
Hence \[B\cup C=\{3,4,5,6,7,8,9,10\}\]
Now we have \[B\cup C=\{3,4,5,6,7,8,9,10\}\] and A = {1, 2, 3, 4, 5}
Hence we have $A\cap \left( B\cup C \right)=\{3,4,5\}$
Let this be called equation (1)
$A\cap \left( B\cup C \right)=\{3,4,5\}...........................\left( 1 \right)$
Now consider $\left( A\cap B \right)\cup \left( A\cap C \right)$
Now first let us find $A\cap B$
Now we have A = {1, 2, 3, 4, 5} and B ={3, 4, 5, 6, 7}
Hence we have \[A\cap B=\{3,4,5\}.............(2)\]
Now consider $A\cap C$
We have A ={1,2,3, 4, 5} and C = {5, 6, 7, 8, 9, 10}
Hence we get $A\cap C=\{5\}............(3)$
Now let us take union of sets obtained in equation (2) and equation (3)
Hence we get
$\left( A\cap B \right)\cup \left( A\cap C \right)=\{3,4,5\}...........(4)$
Hence from equation (4) and equation (1) we can say that $A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$

Note: Note that distributive property of intersection over union is $A\cap \left( B\cup C \right)=\left( A\cap B \right)\cup \left( A\cap C \right)$ and distributive property of union over intersection is $A\cup \left( B\cap C \right)=\left( A\cup B \right)\cap \left( A\cup C \right)$ . We can also verify this property by showing sets in the form of a venn diagram.