Counting Intersections

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Question statement

On an auto-trading website there are $\var{C - F}$ cars listed. Consider this as the set $C$ . Of those $\var{C - F}$ cars in $C$ , $\var{toyota}$ are Toyotas. Consider this as the set $T$ . Similarly, there are $\var{shef}$ cars listed in the Sheffield area. Call this set $S$ .

You should draw a Venn diagram to help you with this question.

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			Name	Type	Generated Value

S	integer	0
F	integer	5
T	integer	12
SF	integer	20
C	integer	40
ST	integer	1
TF	integer	0
SFT	integer	2
shef	integer	23
toyota	integer	15
front	integer	27
shef_toyota	integer	3
shef_front	integer	22
toyota_front	integer	2

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a) In the first part of the question we know that there are $\var{C - F}$ cars listed on the website. We also know that there are $\var{shef}$ cars listed in Sheffield and that $\var{toyota}$ of the total cars listed are Toyotas.

We can use the Inclusion-Exclusion formula to find the cardinality of the intersection: $|S \cap T|$ .

Inclusion-Exclusion: For finite sets $A$ and $B$ , the cardinality of the union is described by the formula $|A \cup B| = |A| + |B| - |A \cap B|$ .

Since we know that $|S \cup T| = \var{C - F}$ , $|S| = \var{shef}$ , and $|T| = \var{toyota}$ , then the size of the intersection is $|S \cap T| = |S| + |T| - |S \cup T| = \var{shef} + \var{toyota} - \var{C - F} = \var{shef + toyota - (C -F)}$ .

b) In this part there are $\var{F}$ more cars listed than before. We need to know how many of the cars listed satisfy all three conditions: in Sheffield, a Toyota, and front-wheel drive. I.e., they are in the intersection $S \cap T \cap F$ .

The Inclusion-Exclusion formula extends to three sets: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$ . This tells us that $|S \cap T \cap F| = |S \cup T \cup F| - |S| - |T| - |F| + |S \cap T| + |S \cap F| + |T \cap F| = \var{C} - \var{shef} - \var{toyota} - \var{front} + \var{shef_toyota} + \var{shef_front} + \var{toyota_front} = \var{SFT}$ .

c) In this part we want the size of the set $(S \cup F) \setminus T$ , i.e., cars which are either in Sheffield or are front-wheel drive, but which are not Toyotas. One way of tackling this is to slowly fill in the sizes of the different regions on a Venn diagram, starting with the middle triple intersection that was worked out in part b). Since we know the size of the pairwise intersections, as well as the single sets, then we can eventually work out that this part needs three numbers to be added together.

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This question is used in the following exam:

FoMCA Formative Maths Test 2425 by Alexander Corner in Fundamentals of Mathematics and Computer Architecture.

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Counting Intersections

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Taxonomy: mathcentre

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Alexander Corner 2 years, 8 months ago

Alexander Corner 2 years, 8 months ago

Alexander Corner 2 years, 8 months ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`