Venn Diagrams

What is a Venn Diagram?

A Venn diagram is a simple way to show how different groups (or sets) relate to each other. It uses circles to represent each group, and where the circles overlap, it shows what the groups have in common.

2-Set Venn Diagrams

Let's begin with two groups:

• Group A: People who like soccer.

• Group B: People who like basketball. In a Venn diagram:

• One circle represents Group A (soccer fans).

• The other circle represents Group B (basketball fans).

• The overlap between the circles shows people who like both soccer and basketball.

  

Key Parts:

• $A \cap B$: The overlap area, where people who like both soccer and basketball are.
• $A \setminus B$: The part of $A$ outside of $B$, where people who only like soccer are.
• $B \setminus A$: The part of $B$ outside of $A$, where people who only like basketball are.
• Outside the circles: Represents people who don't like either soccer or basketball.

3-Set Venn Diagrams

Now, let's add a third group:

• Group C: People who play chess. With three circles:

• The circles represent people who like soccer, basketball, and chess.

• Where two circles overlap, it shows people who like two of these activities.

• Where all three circles overlap, it shows people who like soccer, basketball, and chess.

Simplified View:

• The center where all three circles overlap shows people who like all three activities (soccer, basketball, and chess).
• The overlaps between just two circles show people who like exactly two of the activities.
• The non-overlapping parts of each circle show people who only like one activity.

Solving Word Problems with Venn Diagrams

Let's use a word problem to practice:

Steps to Solve Using a Venn Diagram:

1. Start with the center: Place the 4 students who like all three sports in the middle where all three circles overlap.
2. Work outward:

• 6 students like both soccer and basketball. Since 4 like all three, place the remaining 2 in the overlap between just soccer and basketball.
• 8 students like both hurling and soccer. After accounting for the 4 who like all three, place the remaining 4 in the overlap between just hurling and soccer.
• 5 students like both hurling and basketball. After accounting for the 4 who like all three, place the remaining 1 in the overlap between just hurling and basketball.

3. Fill in the rest:

• For basketball (12 total), subtract the students who also like other sports: $12 - (2 + 1 + 4) = 5$ who only like basketball.
• For soccer (14 total), subtract the students who also like other sports: $14 - (4 + 4 + 2) = 4$ who only like soccer.
• For hurling (15 total), subtract the students who also like other sports: $15 - (4 + 4 + 1) = 6$ who only like hurling.

4. Calculate the outside:

• Add up all the students inside the circles: $4 + 2 + 4 + 1 + 5 + 4 + 6 + 4 = 30$.
• Since the total number of students is 30 and all fit inside the circles, no students like none of these sports.

Why Use Venn Diagrams?

Venn diagrams help you visually organise information and see how groups are related. They're especially useful for problems where you need to figure out how many items (or people) fit into different categories at once.

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