Let's break down and solve the problems shown in the image.

Problem 1: Find the elements of the following sets (a and b).

There is no specific information about sets "a" and "b" in the provided image, so please clarify or provide more details about the sets in question for this part.

Problem 2: Draw the Venn Diagram representing the sets.

The diagram in the image has three sets: A, B, and C. These sets represent different categories, but without specific details for this question, it seems to be a general Venn diagram. However, Problem 3 provides us with data to populate this Venn diagram.

Problem 3: Survey Data and Venn Diagram

Given the survey results:

• 60 students enjoy chess.
• 70 students enjoy board games.
• 75 students enjoy sports.
• 30 students enjoy both board games and sports.
• 18 students enjoy both chess and sports.
• 42 students enjoy both chess and board games.
• 8 students enjoy all three activities.

We are asked to shade the Venn diagram based on this data.

Step-by-step Solution for the Venn Diagram:

1. Let A represent Chess, B represent Board Games, and C represent Sports.

2. Start by filling in the students who enjoy all three activities, which is given as 8 students (i.e., A ∩ B ∩ C = 8).

3. Now, handle the students who enjoy two activities but not all three. We need to subtract the students who enjoy all three from each of these:

   • Students who enjoy both chess and sports: 18 - 8 = 10 (i.e., A ∩ C = 10).
   • Students who enjoy both board games and sports: 30 - 8 = 22 (i.e., B ∩ C = 22).
   • Students who enjoy both chess and board games: 42 - 8 = 34 (i.e., A ∩ B = 34).

4. Next, fill in the number of students who enjoy only one activity:

   • Students who enjoy only chess: 60 - (34 + 10 + 8) = 8.
   • Students who enjoy only board games: 70 - (34 + 22 + 8) = 6.
   • Students who enjoy only sports: 75 - (10 + 22 + 8) = 35.

5. Finally, verify that the total number of students equals 140: $8 \,(\text{only Chess}) + 6 \,(\text{only Board Games}) + 35 \,(\text{only Sports}) + 34 \,(\text{Chess and Board Games}) + 10 \,(\text{Chess and Sports}) + 22 \,(\text{Board Games and Sports}) + 8 \,(\text{all three}) = 140.$ The total is correct.

Now that we've allocated the numbers into the Venn diagram, you can shade each region accordingly.

Would you like me to generate a visual representation of the Venn diagram with the values filled in?

5 Related Questions:

1. How do we calculate the number of students who enjoy only one activity in a Venn diagram?
2. What is the importance of filling in the intersection values before the individual values?
3. Can you find the number of students who enjoy at least two activities?
4. How would the solution change if the total number of students were different?
5. Can we represent more than three sets using a Venn diagram?

Tip:

Always start with the intersection of all sets (where all conditions overlap) when populating a Venn diagram; this helps to avoid double-counting when filling out other regions.