If $ A=\{a,b\},B=(c,d),C=\{d,e\}, $ then $ \{(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)\} $ is equal to
A. $ A\cap \left( B\cup C \right) $
B. $ A\cup \left( B\cap C \right) $
C. $ A\times \left( B\cup C \right) $
D. $ A\times \left( B\cap C \right) $

Hint:We will go through all given options and find out which give us the same answer we are asked to do. While going through options we will need to simplify them using the properties of relation, i.e. $ \cup $ means union, $ '\cap ' $ means intersection and $ '\times ' $ means product.

Complete step by step answer:
Moving ahead with the question in step wise manner;
We need to find out which relation will give the answer $ \{(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)\} $ . By using the given condition, the elements of the set which are $ A=\{a,b\},B=(c,d),C=\{d,e\} $ . So we will go through each option and find out which option will give us the correct relation.
So we can clearly see that every option contains $ B\cup C $ or $ B\cap C $ , so let us first find them; as $ B\cup C $ union of set B with set C which will give us
 $ \begin{align}
  & B\cup C \\
 & (c,d)\cup \{d,e\} \\
 & (c,d,e) \\
\end{align} $
And $ B\cap C $ is intersection of set B with set C, as intersection means common, so it will give us;
 $ \begin{align}
  & B\cap C \\
 & (c,d)\cap \{d,e\} \\
 & (d) \\
\end{align} $
Now moving ahead to check the options we have first $ A\cap \left( B\cup C \right) $ ,
As from above we had find out $ B\cup C $ which is $ (c,d,e) $ and now we need to find out $ A\cap \left( B\cup C \right) $ which is $ \{a,b\}\cap (c,d,e) $ as they do not have any common element so we will get $ \varnothing $ . So this option is wrong.
Let us move to second option which is $ A\cup \left( B\cap C \right) $ , doing it in the similar way we did first option so as from above we had find out $ B\cap C $ which is $ (d) $ and now we need to find out $ A\cup \left( B\cap C \right) $ which is $ \{a,b\}\cap (d) $ as they do not have any common element so we will get $ (a,b,d) $ . So this option is also wrong, as it also did not give us the result we want.
Let us move to third option which is $ A\times \left( B\cup C \right) $ , as here we know $ B\cup C $ which is $ (c,d,e) $ , now let us find out $ A\times \left( B\cup C \right) $ which will be $ \{a,b\}\times (c,d,e) $ so we will get; $ (a,c),(a,d),(a,e),(b,c),(b,d),(b,e) $ so we got 6 elements which match the result we are asked to find out in question. So we can say this option is correct.
Before answering let us also figure out last option, which is $ A\times \left( B\cap C \right) $ , as it is somewhat similar to option ‘c’ so solve it accordingly, as here we know $ B\cap C $ which is $ (d) $ , now let us find out $ A\times \left( B\cap C \right) $ which will be $ \{a,b\}\times (d) $ so we will get; $ (a,d)(b,d) $ which are just 2 elements which consist some of the element similar to asked condition but it does not have all elements, so it is wrong option.

So, the correct answer is “Option C”.

Note: $ A\times B $ is the product of two sets which is done as if set A and set B consist $ \left( a,b \right) $ and $ \left( c,d \right) $ elements, then $ A\times B $ will be simple multiplication of elements that is $ A\times B=\left( a,c \right),\left( a,d \right),\left( b,c \right),\left( b,d \right) $ .