Question

Let A = {10, 15, 20, 25, 30, 35, 40, 45 } 50 , B = { 1, 5,10,15, 20, 30} and C = { 1, 5,15, 20, 35, 45, }. Verify $A\setminus (B\cap C)=(A\setminus B)\cup (A\setminus C)$.

Answer 1

The given sets are $A=\{10,15,20,25,30,35,40,45,50\}$, $B=\{1,5,10,15,20,30\}$ and $C=\{1,5,15,20,35,45\}$

Let us first find $A\setminus (B\cap C)$ as follows:

$A\setminus (B\cap C)=\{10,15,20,25,30,35,40,45,50\}\setminus (\{1,5,10,15,20,30\}\cap \{1,5,15,20,35,45\})$

$=\{10,15,20,25,30,35,40,45,50\}\setminus \{1,5,15,20\}=\{10,25,30,35,40,45,50\}........\left(1\right)$

Now we find $(A\setminus B)\cup (A\setminus C)$as follows:

$(A\setminus B)\cup (A\setminus C)$

$=(\{10,15,20,25,30,35,40,45,50\}\setminus \{1,5,10,15,20,30\})\cup (\{10,15,20,25,30,35,40,45,50\}\setminus$

$\{1,5,15,20,35,45\})$

$=\{25,35,40,45,50\}\cup \{10,25,30,40,50\}=\{10,25,30,35,40,45,50\}........\left(2\right)$

Since equation 1 is equal to equation 2,

Hence, $A\setminus (B\cap C)=(A\setminus B)\cup (A\setminus C)$