Question

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find (B $\cup $ D) $\cap $ (B $\cup $ C).

Answer 1

Hint: As we know that the union of the sets means the collection of all the elements present in the two sets and intersection is the collection of those elements which are in common between the sets. We will apply these definitions between the sets which are considered under union and intersection.

Complete step-by-step answer:
Here we will consider the expression (B $\cup $ D) $\cap $ (B $\cup $ C). After looking at the expression we can clearly observe that the sets which are taken under the union operation are B and D , B and C where B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Now, we will start one by one.
We will first take the expression (B $\cup $ D). Now we will consider the sets B = {4, 5, 6, 7, 8} and D = {10, 11, 12, 13, 14}. Since, we are taking union operations here. So, by the definition of union states the collection of all the elements present in the sets. Therefore, the elements in A and B are 4, 5, 6, 7, 8, 10, 11, 12, 13 and 14. We can write it as,
(B $\cup $ D) = {4, 5, 6, 7, 8} $\cup $ {10, 11, 12, 13, 14}
Therefore, we have (B $\cup $ D) = {4, 5, 6, 7, 8, 10, 11, 12, 13, 14}.
Now we will find the union between the sets for B and C for the expression (B $\cup $ C). For that we have elements B = {4, 5, 6, 7, 8} and C = {7, 8, 9, 10, 11}. And the elements between these are 4, 5, 6, 7, 8, 9, 10 and 11. And, we can write it as,
(B $\cup $ C) = {4, 5, 6, 7, 8} $\cup $ {7, 8, 9, 10, 11}
Therefore, we have (B $\cup $ C) = {4, 5, 6, 7, 8, 9, 10, 11}.
Now we will take the intersection as (B $\cup $ D) $\cap $ (B $\cup $ C). And we will apply the definition of intersection here. That is we will collect common elements. Thus, we have that (B $\cup $ D) $\cap $ (B $\cup $ C) = {4, 5, 6, 7, 8, 10, 11, 12, 13, 14} $\cap $ {4, 5, 6, 7, 8, 9, 10, 11}. Therefore, we get (B $\cup $ D) $\cap $ (B $\cup $ C) = {4, 5, 6, 7, 8, 10, 11}.
Hence, (B $\cup $ D) $\cap $ (B $\cup $ C) = {4, 5, 6, 7, 8, 10, 11}.

Note: As we apply the union operation we are free to collect all the elements that are present between those sets on which the union operation is on. While applying intersection between the sets we consider only those elements which are common between the sets. We must also take care not to repeat any elements in any set, we must represent it only one time.