Question

Consider the following:
$\begin{align}
  & 1.\,\,A\cup (B\cap C)=(A\cap B)\cup (A\cap C) \\
 & 2.\,\,A\cap (B\cup C)=(A\cup B)\cap (A\cup C) \\
\end{align}$
Which of the above is/are correct?
A. \[1\] only.
B. \[2\]only.
C. Both \[1\] and \[2\].
D. Neither \[1\] and \[2\].

Answer 1

Hint: To solve this question we will use the Distribute law property of set theory. We will first take the first equation and see if it is correct or not by using first law of the distributive property. Then to check the second equation we will check use the second law of the Distributive property.



Formula Used:First law of Distributive property of set theory: $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$.
Second law of Distributive property of set theory: \[A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\].



Complete step by step solution:We are given two conditions $A\cup (B\cap C)=(A\cap B)\cup (A\cap C)$ and $\,A\cap (B\cup C)=(A\cup B)\cap (A\cup C)$ and we have to determine which of them is/are correct.
We will first check if $A\cup (B\cap C)=(A\cap B)\cup (A\cap C)$ is correct or not.
According to the first law of the distributive property of set theory for all three sets $A,B$ and $C$ ,$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$. So by this property we can say that given equation $A\cup (B\cap C)=(A\cap B)\cup (A\cap C)$ is not correct.
Now we will check if $\,A\cap (B\cup C)=(A\cup B)\cap (A\cup C)$ is correct or not.
The second law of the distributive property of set theory is that for all three sets $A,B$ and $C$ \[A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\]. With this property we can say that the equation $\,A\cap (B\cup C)=(A\cup B)\cap (A\cup C)$ is not correct.
It means that both the given equations are wrong.



Option ‘D’ is correct


Note: We have used Distributive Law to check both the given condition. This property shows that product and sum will not be changed even if the order of the elements is changed. There are many properties of sets like this which is used in the solution so we must remember these properties to solve these kind of questions. We can also use Venn diagram to check the correctness of the equations.