In a survey,74 out of 88 tourists interviewed said , that had visited at least one of the following continents : Africa, Europe, and South America .Of these , 19 had visited only Europe and Africa,30 Europe and South America, 26 South America and Africa, 10 only Europe 7 only South America and x had visited all the three continents .
(a)Draw a Venn diagram for the information.
(b)Find ,1n[E n A]
2• n[(E u S)' n A]
3.n(A)
(a) The Venn diagram can be divided into three overlapping circles representing Africa, Europe, and South America. The region where all three circles overlap represents the tourists who have visited all three continents. The regions where only two circles overlap represent the tourists who have visited those specific combinations of continents. The regions outside of any circles represent the tourists who have only visited one specific continent.
Here is the Venn diagram:
Africa
/ | \
E E ∩ S S
/ \
Europe Europe ∩ SA
(b) To find 1n[E n A], we look at the overlap between Europe and Africa. From the information given, we know that 19 tourists have visited only Europe and Africa. Therefore, 1n[E n A] = 19.
To find n[(E u S)' n A], we look at the complement of (E u S) intersected with Africa. The complement of (E u S) represents those tourists who have not visited both Europe and South America. From the information given, we know that 10 tourists have visited only Europe and 7 tourists have visited only South America. Therefore, (E u S)' = (10 + 7) = 17. And since 26 tourists have visited Africa and South America, n[(E u S)' n A] = 17.
To find n(A), we add up the number of tourists who have visited Africa in various combinations. From the information given, we know that 19 tourists have visited only Europe and Africa, 26 tourists have visited Africa and South America, and x tourists have visited all three continents. Therefore, n(A) = 19 + 26 + x.
(a) Here is a Venn diagram representing the given information:
```
E SA
= = =========
/ / \ / /
/ / \ / /
/ / \ / /
/ / \ / /
/ / \ / /
/ / X /
/ / / \ /
/ / / \ /
/ / / \ /
/ / / X
============ / / \
A / / \
/ / X
/ / /
/ / /
/ / /
/ / X
/ / /
= X /
E /
= SA
```
(b) Based on the Venn diagram:
1. n[E n A] refers to the number of tourists who have visited both Europe and Africa. From the diagram, we can see that 19 tourists have visited only Europe and Africa, so n[E n A] = 19.
2. n[(E u S)' n A] refers to the number of tourists who have not visited Europe or South America, but have visited Africa. From the diagram, we can see that x tourists have visited all three continents. So, n[(E u S)' n A] = 74 - (10 + 7 + x) = 57 - x.
3. n(A) refers to the number of tourists who have visited Africa. From the diagram, we can see that n(A) = 26 + 19 + 7 + x = 52 + x.