Question

Which of the following represent the distributive law set?
A. $A \cup (B \cap C) = (A \cap B) \cup (A \cap C)$
B. $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
C. $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
D. $A \cap (B \cup C) = (A \cup B) \cap (A \cup C)$

Answer 1

Hint:
According to the distributive law of sets, the given two equations are always true
\[A\; \cup \left( {B\; \cap \;C} \right)\; = (A\; \cup \;B)\; \cap (A\; \cup \;C)\]
\[A\; \cap (B\; \cup \;C)\; = \left( {A\; \cap \;B} \right)\; \cup \left( {A\; \cap \;C} \right)\]
Hence, we will compare these two equations with the given choices to get to the final answer.

Complete step by step solution:
According to the distributive law of sets, if we are given three sets A, B, and C, then the given two relations of these
 \[A\; \cup \left( {B\; \cap \;C} \right)\; = (A\; \cup \;B)\; \cap (A\; \cup \;C)\] … (1)
 \[A\; \cap (B\; \cup \;C)\; = \left( {A\; \cap \;B} \right)\; \cup \left( {A\; \cap \;C} \right)\] … (2)
Now when we compare these two equations with the choices given

Option1: \[A\; \cap (B\; \cup \;C)\; = \left( {A\; \cap \;B} \right)\; \cup \left( {A\; \cap \;C} \right)\]
After comparing Option1 with both the equations, we can say that
Option1 does not match any of the two equations, Hence it is eliminated.

Option2: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
After comparing Option2 with both the equations, we can say that
Option2 matches (2), Hence it is selected

Option3: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
After comparing Option3 with both the equations, we can say that
Option3 matches (1), Hence it is selected

Option4: $A \cap (B \cup C) = (A \cup B) \cap (A \cup C)$
After comparing Option4 with both the equations, we can say that
Option4 does not match any of the two equations, Hence it is eliminated

Hence, the correct answer is option B and C.

Note:
In the questions which have multiple choices correct we must check all the choices given before arriving at the answer, If we stop after finding only one correct choice, our answer may be correct but there are high chances that it would be incomplete and hence wrong.