Venn diagrams & tables

Venn diagrams

A Venn diagram describes how sets of things can be categorized. Usually, there are two characteristics by which all things in the set are categorized, although the technique can extend to any number of categorizations. The catch is that some parts of the set can be within multiple categories; this is called the intersection of the Venn diagram.

For example, the Venn diagram below represents a group of students in a graduating class. It categorizes the students by two traits: if they participated in extracurricular activities, and if they had a high GPA. Like most diagrams on the GRE, Venn diagrams are typically not drawn to scale.

Some students might be in only one group (activities or GPA), some might be in both groups (activities and GPA), and some might not be in either group (not activities and not GPA).

Looking at our diagram, we can see that 5 students participated in activities and had a high GPA. An additional 15 students participated in activities but did not have a high GPA, and another 10 students had a high GPA but didn’t participate in activities. To get the total number of students, sum up all the numbers, ensuring any overlap isn’t double-counted. In total, we have $15+5+10=30$ students, of which $15+5=20$ participated in activities, and $5+10=15$ had a high GPA.

GRE problems relating to Venn diagrams typically give you a few numbers to explain part of the scenario, and then it’s your job to figure out the rest. You might only be told the total and the intersection, or the total in each category but not the overlap. In these cases, you’ll just need to solve for the missing part.

Range of values Venn diagram questions

More complicated problems might give you even less information. It’s possible that you might not be able to solve for any exact values in the Venn diagram, and instead have to determine the range of values that could fit. Working with a range-based Venn diagram question is like a normal range-based question. Determine the lowest and highest numbers that could fit, i.e., the two situations when the Venn diagram circles have the smallest and largest overlap. These numbers are called the extremes, and they give you the range of possible values that work, including the extremes themselves.

Here’s an example:

A cafe serves coffee and pastries to its customers. There were a total of 70 customers at the cafe today. If there were 5 more customers who ordered only pastries than the number of customers who only ordered both pastries and coffee, how many could have ordered just coffee?

Select all that apply.
A. 0
B. 2
C. 33
D. 64
E. 66

Let’s start by imagining the smallest extreme for the number of customers that ordered only both coffee and pastries. It’s possible that nobody ordered both, so the smallest overlap is $0$, which would mean that 5 people ordered pastries only. This leaves $70-5=65$ people ordering coffee only. Our Venn diagram would look like this:

The opposite extreme has the largest overlap. We just need to ensure that the number of people ordering pastries only is 5 more than the number of people ordering both coffee and pastries.

You could figure it out by trial and error filling in the circles, or alternatively, set up a simple equation to solve:

$$\begin{array}{rl}(x+5)+x& =70\\ 2x+5& =70\\ 2x& =65\\ x& =32.5\end{array}$$

We can’t have half a person, so the overlap must be 32, with one person being a rebel and drinking coffee only.

So, the two possible extremes for the number of only coffee drinkers are 1 and 65, indicating that the correct answers to this question are in the range $1\le x\le 65$.

However, some choices in this range won’t be possible because we can’t have half a person. For instance, if we had 2 coffee-only drinkers, by the rules of the question, we would need to have 36.5 pastry-only people and 31.5 coffee-and-pastry people. We would need to subtract another half a person from the pastry-only people and the coffee-and-pastry people, leaving us with 3 coffee-only, 36 pastry-only, and 31 coffee-and-pastry. Extrapolating from this logic, it is not possible to have an even number of coffee-only people, so the choices B. 2 and D. 64 are incorrect, even though they fall within the range.

A. 0
B. 2
C. 33 - CORRECT
D. 64
E. 66

Union and intersection

Venn diagrams are great tools to learn about the union and intersection of sets.

The intersection of two sets A and B will usually have fewer elements than the union of A and B. If the sets are identical, their unions and intersections will also be identical: $A=B=A\cup B=A\cap B$. It’s impossible for the intersection of sets to have more elements than the union of those sets.

In the Venn diagram above, $A\cup B$ is equivalent to $A+B+C$. The intersection $A\cap B$ is equivalent to just $C$. Another tip to help remember the symbols is to imagine the closed, upside down, $\cap$ symbol as the closed, landlocked section of the Venn diagram. The upward, more open, $\cup$ symbol includes everything in the Venn diagram.

Try this relatively simple example question:

List $A$: [ 3, 6, 9, 18, 23 ]
List $B$: [ 2, 6, 9, 10, 18 ]
List $C$: [ 1, 3, 5 ]

Quantity A: The sum of the elements of $A\cap B$
Quantity B: The sum of the elements of $B\cup C$

You should have all the knowledge you need to solve this one!

Start by listing out the elements in the quantities.

The elements in Quantity A, $A\cap B$, are the ones that are shared in both List $A$ and List $B$: $[6,9,18]$. Their sum is $6+9+18=33$.

The elements in Quantity B, $B\cup C$, are the ones that are present in either List $B$ or List $C$: $[1,2,3,5,6,9,10,18]$. Their sum is $1+2+3+5+6+9+10+18=54$.

We’ve solved the question by listing out all the values. Setting back and thinking logically, Quantity B will always be equal to or greater than Quantity A regardless of the sets’ values. Quantity A’s intersection is a subset of List $B$, and Quantity B’s union includes the entirety of List $B$ plus the elements in List $C$.

Tables

Another common question type will describe a set of items that can be categorized in two or more unique ways. This situation differs from a Venn diagram because each item has an identity in each category. For example, let’s consider a deck of cards. The suit color of any card will always be black $♣♠$ or red $♡♢$. Additionally, each card may also be categorized as a face card (i.e. JQKA) or a number card (1-9). Here is a table that describes the number of cards in each subset:

See how the sum of each row or column adds up to the totals? The bottom right number describes the total for the entire group.

In this example, all the information was already known. Sometimes a problem will only offer a few pieces of information, and similar to the game Sudoku, you should use the information present to solve for the missing spaces. Start with a column that is only missing one value, and use simple addition/subtraction to find it. You’ll be able to work your way through the table until all the blank spaces are filled in.