ICSE Class 8 Maths Chapter 38 Venn Diagrams

Chapter 38: Venn-Diagrams

Review

Venn-diagram is the most commonly used pictorial representation of sets. This idea was first developed by John Venn, an English mathematician, that is why the figures (geometrical figures) used in this type of representation are called Venn-diagrams.

In fact, in a Venn-diagram, a closed curve (figure) represents a set and the interior points within this closed curve represent the elements of the set.

In a Venn-diagram, the universal set is represented by a rectangle and all other sets under consideration by circles or ovals within the rectangle.

Use of Venn-Diagrams to Show the Relationship Between the Sets

1. The diagram shows two disjoint sets A and B.

2. The diagram shows A and B are joint or overlapping sets.

3. (i) \(A \subset B\)

(ii) \(B \subset A\)

4. The shaded portions in the following figures show \(A \cup B\).

(i) Overlapping sets with both circles shaded

(ii) Disjoint sets with both circles shaded

(iii) Single circle fully shaded

5. The shaded portions in the following figures show \(A \cap B\).

(i) Overlapping sets with intersection shaded

(ii) Single circle with horizontal lines shaded inside another circle

(iii) Two disjoint circles with no shading

6. The shaded portions in the following figures show \(A - B\).

(i) Left portion of overlapping sets shaded

(ii) Left circle completely shaded while right is empty

(iii) Left circle shaded, right circle empty with no overlap

7. The shaded portions in the following figures show \(B - A\).

(i) Right portion of overlapping sets shaded

(ii) Right circle shaded with left empty

(iii) Right circle shaded inside larger circle

Example 1

For two overlapping sets A and B, draw Venn-diagrams to represent the following sets:

(i) \((A \cap B)'\)

(ii) \((A \cup B)'\)

(iii) \(A' \cap B\)

Solution

(i) Since, \(A \cap B\) = overlapping region, therefore \((A \cap B)' = \) the region of universal set, which is not in \(A \cap B\). The diagram shows horizontal shading in all areas except the central intersection.

(ii) Since, \(A \cup B\) = both circles combined, therefore \((A \cup B)' = \) regions outside both circles. The diagram shows horizontal shading only in the area outside both circles.

(iii) Since, \(A' = \) everything except A (shown with horizontal shading), therefore \(A' \cap B = \) the region in both A' and B. The diagram shows the right portion of B shaded with horizontal lines.

Example 2

Use Venn-diagrams to prove that: \((A \cap B)' = A' \cup B'\).

Solution

Consider A and B as two overlapping sets.

Since, \(A \cap B = \) central overlapping region, therefore \((A \cap B)' = \) all regions except the central intersection. This is labeled as diagram I.

Since, \(A' = \) everything outside circle A (shown with horizontal shading) and \(B' = \) everything outside circle B (shown with vertical shading), therefore \(A' \cup B' = \) all regions except the central intersection. This is labeled as diagram II.

Since the shaded portions in diagrams I and II represent the same region, therefore \((A \cap B)' = A' \cup B'\). Hence proved.

Example 3

Given: \(\xi = \{x : x \in \mathbb{N}, 12 \le x < 20\}\), \(A = \{x : x \text{ is divisible by } 3\}\) and \(B = \{12, 14, 15, 16\}\). Draw a Venn-diagram to show the relationship between the given sets.

Solution

Given: \(\xi = \{12, 13, 14, 15, 16, 17, 18, 19\}\), \(A = \{12, 15, 18\}\) and \(B = \{12, 14, 15, 16\}\).

Since the sets A and B are overlapping sets, therefore, the Venn-diagram should be as drawn alongside.

The diagram shows: Circle A contains 18 on the left, 12 and 15 in the overlapping region, circle B contains 16 and 14 on the right, and outside both circles are 13 and 19.

Teacher's Note

Venn diagrams help us organize information visually, just like organizing your closet by putting similar items together - clothes that are both blue AND cotton go in one spot, while items that are blue OR cotton go in different sections.

Test Yourself

Using the adjoining figure, find the following sets:

1. \(A \cap C = \) ..................

2. \(B \cup C = \) ..................

3. \(A \cup B = \) .................. and \((A \cup B) \cap C = \) ..................

4. \(A \cup C = \) .................. and \((A \cup C)' = \) ..................

5. \(C - A = \) ..................

6. \(B - C = \) ..................

7. \(C - B = \) .................. and\((C - B) \cup A = \) ..................

8. \((A \cup C) - B = \) ..................

The figure shows three overlapping circles labeled A, B, and C with regions containing numbers: 7, 15, 13 in the upper section; 4, 10, 5 in the middle areas; 12, 11 in the lower section; and 6, 9 outside with 14 on the far left and 8 on the far right.

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