Venn diagrams & tables

Venn diagrams

A Venn diagram shows how a group of items can be sorted into categories (called sets). Most GRE Venn diagrams use two categories, but the idea can extend to more. The key feature is that an item can belong to more than one category. The region where categories overlap is called the intersection.

For example, the Venn diagram below represents a graduating class. It sorts students by two traits:

• whether they participated in extracurricular activities
• whether they had a high GPA

Like most GRE diagrams, Venn diagrams usually aren’t drawn to scale.

In a two-circle Venn diagram:

• some students are in only one group (activities or GPA)
• some are in both groups (activities and GPA)
• some are in neither group (not activities and not GPA)

From the diagram:

• 5 students participated in activities and had a high GPA (the overlap)
• 15 students participated in activities but did not have a high GPA
• 10 students had a high GPA but did not participate in activities

To find the total number of students, add all regions, making sure the overlap is counted only once:

• Total: $15+5+10=30$
• In activities: $15+5=20$
• High GPA: $5+10=15$

GRE Venn diagram problems typically give you some of these numbers and ask you to find the rest. For example, you might be given the total and the intersection, or the totals in each circle but not the overlap. In those cases, you solve for the missing region(s).

Range of values Venn diagram questions

Some problems give so little information that you can’t determine exact values. Instead, you have to find the range of values that could work.

To do this, find the two extremes:

• the smallest value that could fit the conditions
• the largest value that could fit the conditions

These extremes define the full range of possible values (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 at least 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

Start with the smallest possible overlap (customers who ordered both coffee and pastries). The smallest overlap is $0$. Then the condition “at least 5 more pastries-only than both” means pastries-only could be $5$. That leaves:

• coffee-only: $70-5=65$

Your Venn diagram would look like this:

Now look for the opposite extreme: the largest possible overlap. To maximize the overlap, you still have to satisfy the condition that pastries-only is 5 more than the overlap.

You can find this by trial and error, or by writing an equation. Let $x$ be the number of customers who ordered both coffee and pastries. Then pastries-only is $x+5$, and (for the extreme case) coffee-only is $0$. The total is 70:

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

You can’t have half a person, so the overlap can’t be 32.5. The closest whole-number overlap that still keeps the total at 70 is 32, which forces 1 person into the coffee-only region.

So the two extremes for coffee-only are:

• minimum: 1
• maximum: 65

That suggests the range $1\le x\le 65$ for the number of coffee-only customers.

However, not every number in that range is actually possible because all regions must be whole numbers. For example, if coffee-only were 2, then the remaining 68 customers would have to be split into pastries-only and both, with pastries-only 5 more than both. That would require 36.5 pastries-only and 31.5 both, which isn’t possible.

Following this logic, an even number of coffee-only customers won’t work here, so choices B. 2 and D. 64 are not possible even though they fall within the numerical range.

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

Union and intersection

Venn diagrams are a useful way to visualize the union and intersection of sets.

Definitions

Union

The combination of two or more sets. The union of sets A and B, represented as $A\cup B$, includes all the elements in A and all the elements in B. The symbol for union $\cup$ looks like the letter u.

Intersection

The overlap of two or more sets. The intersection of sets A and B, represented as $A\cap B$, includes only the elements present in both A and B. The symbol for intersection $\cap$ looks like the letter n.

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, then their union and intersection are also identical: $A=B=A\cup B=A\cap B$. The intersection of sets can’t have more elements than the union of those sets.

In the Venn diagram above:

• $A\cup B$ includes everything in either circle, so it’s equivalent to $A+B+C$.
• $A\cap B$ includes only the overlap, so it’s equivalent to $C$.

A memory tip: the closed, upside-down $\cap$ symbol matches the “closed-in” overlap region, while the more open $\cup$ symbol matches the idea of including everything.

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 the elements in each quantity.

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

This problem is easiest to solve by listing values, but it also makes sense conceptually: Quantity A is an intersection (a subset of List $B$), while Quantity B is a union (all of List $B$, plus anything extra from List $C$). So Quantity B will always be at least as large as the sum of List $B$, while Quantity A can only include elements that are shared.

Tables

Another common question type describes a group of items that can be categorized in two or more separate ways, where every item belongs to exactly one category in each dimension. This differs from a Venn diagram, where an item can belong to multiple categories at once.

For example, consider a deck of cards:

• Every card is either black $♣♠$ or red $♡♢$.
• Every card is either a face card (i.e. JQKA) or a number card (1-9).

Here is a table showing the number of cards in each subset:

	Black suit	Red suit	Total
Face card	8	8	16
Number card	18	18	36
Total	26	26	52

Notice that each row and column adds to its total. The bottom-right value is the total number of items in the entire group.

In this example, all the values are already filled in. On the GRE, you may be given only some entries. Similar to the game Sudoku, you use the totals to fill in the missing values:

• Start with a row or column that is missing only one value.
• Use addition or subtraction to find that value.
• Repeat until the table is complete.