Unit 1 Section 5 : Logic Problems and Venn Diagrams

Venn diagrams can be helpful in solving logic problems.

Example Questions

In a class there are:

• 8 students who play football and hockey
• 7 students who do not play football or hockey
• 13 students who play hockey
• 19 students who play football

We can draw a venn diagram to show the numbers playing football (F) or hockey (H). We can put in the numbers for the first two facts straight away, as seen on the left. The 8 students who play both hockey and football go in the intersection because they need to be in both circles. The 7 students who don't play either sport go on the outside because they shouldn't be in either circle.

We have to be careful with the other two facts. There are 13 students who play hockey, so the numbers in the hockey circle should add up to 13. We already have 8 in the intersection, so there must be 5 who play hockey but not football. In the same way there are 19 students who play football so the number who play football but not hockey must be 19 – 8 = 11. To find out the number of students in the class we add up all the sections: 11 + 8 + 5 + 7 = 31 There are 31 students in the class.

Question 2

In a class there are 30 students.

• 21 students like Maths
• 16 students like English
• 6 students don't like Maths or English

	We can draw a venn diagram to show the numbers who like Maths (M) or English (E), but this time we can only put in a number for one of the facts straight away. The 6 students who don't like either subject go on the outside because they shouldn't be in either circle. We know the total in the Maths circle needs to be 21 but we can't put this in because we don't know how many should go in the intersection (if they like both subjects) and how many should go on the left (if they only like Maths).
	We know there are 30 students in the class, and if there are 6 students outside the circles then the other three sections must add up to 24. We know there are 21 students who like Maths, so the middle and left section must add up to 21. This leaves 3 on the right because 24 – 21 = 3.
	There are 16 students who like English so the two parts of the English circle should add up to 16, so we can find the number in the intersection by doing 16 – 3 = 13. There are 21 students who like Maths, and 21 – 13 = 8, so the number who like Maths but not English must be 8. If we check all four facts we were given, we can now see they are all true. There are 13 students who like both Maths and English.

Practice Question

There are 18 cars on a garage forecourt. 12 cars are diesels. 5 cars are automatics. 3 cars are automatic diesels. To find out how many cars are not automatic and not a diesel, work through the stages below.

(a) Where do the 3 automatic diesels get marked on the diagram?

(b) If there are 12 diesels, how many diesels are not automatic?

(c) How many of the automatic cars are not diesels?

(d) How many cars are not automatic and not a diesel?

Exercises

It may help to have a pencil and paper handy so you can sketch venn diagrams to help you answer the questions.

Question 10
All the members of a group of 30 teenagers belong to at least one club.
There are 3 clubs, chess, drama and art.

• 6 of the teenagers belong to only the art club.
• 5 of the teenagers belong to all 3 clubs.
• 2 of the teenagers belong to the chess and art clubs but not to the drama club.
• 15 of the teenagers belong to the art club.
• 2 of the teenagers belong only to the chess club.
• 3 of the teenagers belong only to the drama club.

(a) How many of the group do chess and drama but not art?
(b) How many of the group belong to the chess club?