Let A = {2, 3, 4, 5}, B = {3, 6, 9}, and C = {5, 6, 7, 8}. Find A ∩ (B ∪ C) and B ∪ (A ∩ C).

Understand the Problem

The question is asking to find the intersection and union of three sets A, B, and C, where specific elements are provided for each set. The tasks involve calculating A intersection (B union C) and B union (A intersection C).

Answer

$A \cap (B \cup C) = \{3, 5\}, \, B \cup (A \cap C) = \{3, 5, 6, 9\}$

Steps to Solve

Identify Sets
We have the sets defined as:
$$ A = {2, 3, 4, 5} $$
$$ B = {3, 6, 9} $$
$$ C = {5, 6, 7, 8} $$
Find B Union C
We need to find the union of sets B and C.
The union of two sets A and B, denoted as $A \cup B$, is the set of elements that are in A, or B, or both.
So,
$$ B \cup C = {3, 6, 9} \cup {5, 6, 7, 8} = {3, 5, 6, 7, 8, 9} $$
Find A Intersection (B Union C)
Now we will find the intersection of set A with the union we just calculated.
The intersection of two sets A and B, denoted as $A \cap B$, is the set of elements that are in both A and B.
Thus,
$$ A \cap (B \cup C) = {2, 3, 4, 5} \cap {3, 5, 6, 7, 8, 9} = {3, 5} $$
Find A Intersection C
Next, we find the intersection of sets A and C:
$$ A \cap C = {2, 3, 4, 5} \cap {5, 6, 7, 8} = {5} $$
Find B Union (A Intersection C)
Finally, we find the union of set B and the intersection of A and C:
$$ B \cup (A \cap C) = {3, 6, 9} \cup {5} = {3, 5, 6, 9} $$

The results are:
$$ A \cap (B \cup C) = {3, 5} $$
$$ B \cup (A \cap C) = {3, 5, 6, 9} $$

More Information

The intersection of sets finds the common elements, while the union combines all distinct elements from the involved sets.

Tips

Confusing intersection with union: Remember that intersection ($A \cap B$) finds common elements, while union ($A \cup B$) combines all distinct elements.
Neglecting to include all elements in the union step: Ensure that no duplicates are included in the union.

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