This is what you need to know from SET THEORY for your McKinsey PST test:
00:30 Example 01
03:27 Detailed explanation of the solution
– 03:27 Step 01 – Start with what you know
– 04:21 Step 02 – Tackle the unknown subsets one by one
– 05:20 Step 03 – Find the answer
06:00 Euler Diagram solution algorithm
06:31 Example 02
Euler diagrams is the topic of the 12th theory session of the problem-solving test course. We will learn how to calculate intersections and unions of various sets and also find out why this knowledge is important for the McKinsey PST. We will kick off with exploring hypothetic class hypothetic business school.
You might want to pause the video now as always and try to tackle the question on your own. Based on the information above how many students in the class of 2020 at the Niqua Business School have both consulting and industry experience?
From the text we know that the class consist of 100 students 75 of them have consulting background and 50 have industry background and 20 students have neither consulting nor industry background. And the question is how many of these students have both consulting and industry background?
The best way to tackle this question is by using a Euler diagram. Let’s that’s draw it here: the total number of students is 100; those with consulting background 75 people; those with industry background 50 people; those without consulting an industry background already here 20 people. Then the union of this sets: people who have either consulting or industry background is 80 people. And we want to find the number of people who have both consulting and industry background?
This is these subsets here.
How do we do that? We need to find these segments first?
And then we will be able to find the answer, the intersection of the sets, by subtracting the number of people with consulting background only, these part here, from the total number of people with consulting background.
So, the number of people with consulting background only I would call it A and not B; where this is A and B is 80 minus 50.
These guys here equal 30.
And then the number of people with consulting and industry background is 75, minus these are here, A without B 30 equals 45. And this is our answer, option C.
Let us consider the solution of this problem more slowly.
Euler diagram is a way of visualizing sets their unions and intersections.
This visualization will help us establish relations among sets and subsets and move from known sides of sets to the unknown. In the example above, we start with the general set: the whole class. Hundred students, we call it S. Then we draw a subset A: all x consultants which 75 students is. A should be within S, because it’s a subset of S.
Then draw a subset B:
All 50 students from the industry. Notice that A and B must intersect, because there might be students with both consulting and industry experience.
Also, A and B lie within S.
At the next step consider the set within S but outside of a and B.
It is called complementary to the set A or B.
We know that its size is 20 people.
Now we are ready to compute some unknown sizes of a sets. Set A or B: students with consulting or industry background, or both.
Its size is equal to S minus the size of its complement. Namely: hundred minus 20 equals 80 people. The set A and not B has the set B as a complement for the set A or B, whose size is 80. Therefore, the size of A and B is 18 minus 50, the size of B or 30 people.
30 people have consulting background but not industry background. Symmetrically B and not A has the size of 80 minus 75 the size of A or five people.
Five people have industry background, but not consulting background. Ok, now we’re ready to find the number of people who have both consulting and industry background. A and B is a complementary A for B and not a and a complementary set for A and not B, so its size is 50 minus 5, or equivalently 75 minus 30, or 80 minus 30 minus 5 which is forty-five people.
This is the desired result.
The Euler diagram in this example help us understand the relations among sets and figure out how to mix and match them to ger where we need.
This is why the diagram is useful.
There is no simple one size fits all algorithm for solving set questions, as we’ll see in the next example.
We have to come up with a solution using the diagrams visualization.
However, a very generic algorithm might look as follows.
Draw sets as rectangles and circles, pay attention to their intersections.
Then assign known values to sets and then calculate the unknowns one by one by means of operations with complements unions and intersections.
The connections between pictures and corresponding math operations are shown on the right.
Here’s a more complicated example. Pause the video now and try to solve the question on your own, and after that, continue watching. If three out of thirty-eight associates work long hours and do not have time for sports at all, how many associates at ZeroSum consulting are active in all three mentioned sports activities? From the text we learned that there are 38 associates, 16 of them play basketball, 18 play soccer and 17 are surfers. And then, several consultants are active in exactly two sports activities: four in surfing and basketball; three in soccer and basketball; and then five in surfing and soccer.
Let’s draw a Euler diagram and try to tackle the question using it.
The total number of people is 38 and then, there are three intersecting sets B, people who play basketball; soccer, people who play soccer; and then surfing, people who do surfing.
Then there are 16 people here, 18 people here, and 17 people here. And we also know that the number of people in this segment, basketball and surfing only, is 4; the number of people in basketball and soccer only is three.
And then the number of people in soccer and surfing here is five.
And we need to find the size of these small set here.
We also know that there are three people who don’t do sports at all.
They are here. So, the first step would be to calculate the union of these three sets, because three is the size of the complement to the set, we are computing.
Now we need to calculate the intersection of these three sets, and this would be possible if we expressed the size of basketball or soccer or surfing in different terms, because equivalently we can write 16 plus 18, plus 17, which is the sum of these three sets, but then we also had some double counting because we counted this segment twice, this segment twice, this segment twice and then this segment here three times.
So, we need to subtract four, we need to subtract three, we need subtract five and then, and we need to subtract x the size of our desired set two times, because in this calculation here, it was included three times and we only need one.
So, there are two people who are active going over three sports activities.
And this is our answer.
Let us synthesize what we have learned today. Euler diagram are great tool for visualizing relations among sets, which is helpful in solving questions about sets.
We need to draw all sets, pay attention to compliments, unions and intersections.
Then we write down the sizes we know. Then we start computing the sizes we do not know and the visualization helps us decide how to do that.
After some struggle, we arrive at the answer.
That’s it for today.
If you have any questions I’ll be happy to read it answer them in the comments below.
Thanks for your time.
And then to next session.