Let A, B, C’ be three sets as shown in the following Venn diagram. For each of the following sets, draw a Venn diagram and shade the area representing the given set. a. AU BUC b. An Bnc C. AU (BNC) d. A – (BNC) e. AU (BNC)
The Correct Answer and Explanation is :
Let’s break down each of the sets given, describing the notation and how they would appear in a Venn diagram.
- a. ( A \cup B \cup C ):
- This set represents the union of sets ( A ), ( B ), and ( C ). The union symbol (( \cup )) means combining all the elements from ( A ), ( B ), and ( C ) into one set, so every part of all three sets should be shaded. Essentially, the entire area of the three overlapping circles should be shaded, including the intersections of ( A ) with ( B ), ( B ) with ( C ), and ( A ) with ( C ), as well as the individual parts of each circle.
- b. ( A \cap B \cap C ):
- This represents the intersection of all three sets ( A ), ( B ), and ( C ). The intersection symbol (( \cap )) means only the elements that are common to all three sets. In this Venn diagram, the shaded area would only be the part where all three circles overlap, at the very center where ( A ), ( B ), and ( C ) meet.
- c. ( A \cup (B \cap C) ):
- This set represents the union of set ( A ) and the intersection of ( B ) and ( C ). To shade this in a Venn diagram, shade the entire area of ( A ) and also shade the intersection of ( B ) and ( C ) (i.e., where ( B ) and ( C ) overlap). The parts that are in ( B ) and ( C ), but not in ( A ), should also be shaded.
- d. ( A – (B \cap C) ):
- The subtraction of ( B \cap C ) from ( A ) means you exclude the area where ( B ) and ( C ) overlap from the set ( A ). In the Venn diagram, shade all of ( A ) except the part where ( B ) and ( C ) intersect. This represents all of ( A ), but excluding the portion that is shared with ( B ) and ( C ).
- e. ( A \cup (B \cap C) ):
- This is exactly the same as option (c), so the shading remains the same. Shade all of ( A ) and the area where ( B ) and ( C ) intersect.
Explanation of Notation:
- Union (( \cup )): The union of sets combines all elements in the sets involved, without duplication.
- Intersection (( \cap )): The intersection of sets consists of elements that are common to all sets involved.
- Difference (( – )): The difference between two sets removes elements of the second set from the first.
In summary, each part of the Venn diagram corresponds to specific relationships between the sets ( A ), ( B ), and ( C ). By understanding these set operations and their graphical representation in Venn diagrams, we can visualize complex relationships and operations between sets.