ISEE Upper Level

Set Theory

Set notation, Venn diagrams, union, intersection, and complement — excludes permutation/combination counting (see counting-combinations) and pure probability (see probability)

Generate Unlimited Practice Questions

Start Free Practice

Learn This Topic

Imagine you and your best friend are ordering a pizza. You want pepperoni, and your friend wants mushrooms. To make you both happy, you order a half-pepperoni, half-mushroom pizza, but you ask the chef to put extra cheese all over the whole thing! 🍕

In math, we call groups of things "Sets." Your pizza toppings are a set, and your friend's toppings are a set. Set Theory is just a fun way of organizing these groups to see what they have in common.

On the ISEE, you will see a lot of "Venn diagrams." These are those cool overlapping circles. - The Intersection is the overlapping middle part. It’s what BOTH sets share (like the extra cheese you both wanted). - The Union is EVERYTHING in both circles combined (pepperoni, mushrooms, and extra cheese).

The ISEE Quantitative Reasoning section loves to test if you can figure out how many kids play sports, or how many pets are in a store, by using these overlapping groups. The biggest trick is to make sure you don't accidentally count the same thing twice! Let's learn the secret formula to solve these puzzles like a pro. 🕵️‍♂️

Key Formula

The "Magic Overlap" Formula for two sets:

T o t a l = G r o u p A + G r o u pB - B o t h + N e i t h er

. Always subtract the "Both" group so you don't double-count them!

Practice Questions

4 practice questions for ISEE Upper Level

Q1 Hard

Given the universal set

U = {x \in Z ∣ 0 \leq x \leq 10}

and sets

A = {1, 3, 5, 7, 9}

B = {2, 3, 5, 7}

, and

C = {1, 2, 4, 8, 9, 10}

. Which of the following sets is equal to

(A \cap B^{'}) \cup C

{1, 2, 4, 8, 9, 10}

{1, 2, 3, 4, 5, 7, 8, 9, 10}

{1, 9}

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Show Solution

First, identify the universal set $U$ .

$U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ .
Next, find the complement of set B, denoted $B^{'}$ . $B^{'}$ contains all elements in $U$ that are not in $B$ .
$B = {2, 3, 5, 7}$ .
$B^{'} = {0, 1, 4, 6, 8, 9, 10}$ .
Then, find the intersection of set A and $B^{'}$ , denoted $A \cap B^{'}$ . This set contains elements common to both A and $B^{'}$ .
$A = {1, 3, 5, 7, 9}$ .
$B^{'} = {0, 1, 4, 6, 8, 9, 10}$ .
$A \cap B^{'} = {1, 9}$ .
Finally, find the union of $(A \cap B^{'})$ and set C, denoted $(A \cap B^{'}) \cup C$ . This set contains all elements from $A \cap B^{'}$ or C, without duplication.
$A \cap B^{'} = {1, 9}$ .
$C = {1, 2, 4, 8, 9, 10}$ .
$(A \cap B^{'}) \cup C = {1, 2, 4, 8, 9, 10}$ .
Therefore, the correct set is ${1, 2, 4, 8, 9, 10}$ .

Answer: A

Q2 Hard

A survey of 100 high school students revealed their participation in three school clubs: Math Club (M), Science Club (S), and Debate Club (D). The results were:
- 40 students are in the Math Club.
- 35 students are in the Science Club.
- 30 students are in the Debate Club.
- 15 students are in both the Math Club and the Science Club.
- 12 students are in both the Science Club and the Debate Club.
- 10 students are in both the Math Club and the Debate Club.
- 5 students are in all three clubs.
How many students are in exactly one club?

A 40

B 42

C 44

D 46

Show Solution

To solve this, we can use a Venn diagram approach. Let's break down the numbers for each region:
1. Students in all three clubs ( $M \cap S \cap D$ ):

Given: 5 students.
2. Students in exactly two clubs (but not all three):

• Math and Science only ( $M \cap S$ only): $(M \cap S)$ - $(M \cap S \cap D)$ = 15 - 5 = 10 students.
• Science and Debate only ( $S \cap D$ only): $(S \cap D)$ - $(M \cap S \cap D)$ = 12 - 5 = 7 students.
• Math and Debate only ( $M \cap D$ only): $(M \cap D)$ - $(M \cap S \cap D)$ = 10 - 5 = 5 students.
3. Students in exactly one club:

• Math Club only (M only): $M$ - (students in $M \cap S$ only) - (students in $M \cap D$ only) - (students in all three)
$M$ only = 40 - (10 + 5 + 5) = 40 - 20 = 20 students.
• Science Club only (S only): $S$ - (students in $M \cap S$ only) - (students in $S \cap D$ only) - (students in all three)
$S$ only = 35 - (10 + 7 + 5) = 35 - 22 = 13 students.
• Debate Club only (D only): $D$ - (students in $S \cap D$ only) - (students in $M \cap D$ only) - (students in all three)
$D$ only = 30 - (7 + 5 + 5) = 30 - 17 = 13 students.
4. Total students in exactly one club:

Sum the students in M only, S only, and D only.
20 (M only) + 13 (S only) + 13 (D only) = 46 students.
Therefore, 46 students are in exactly one club.

Answer: D

Q3 Hard

Let the universal set

U = {x \in Z ∣ 0 \leq x \leq 15}

.
Let

P = {x ∣ x is a prime number in U}

.
Let

Q = {x ∣ x is an even number in U}

.
Which of the following sets is equal to

P \cup Q^{'}

{1, 2, 3, 5, 7, 9, 11, 13, 15}

{0, 1, 2, 3, 5, 7, 9, 11, 13, 15}

{2, 3, 5, 7, 11, 13}

{1, 3, 5, 7, 9, 11, 13, 15}

Show Solution

First, identify the universal set $U$ .

$U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}$ .
Next, identify the set P (prime numbers in U). Remember 0 and 1 are not prime.
$P = {2, 3, 5, 7, 11, 13}$ .
Next, identify the set Q (even numbers in U).
$Q = {0, 2, 4, 6, 8, 10, 12, 14}$ .
Now, find the complement of Q, denoted $Q^{'}$ . $Q^{'}$ contains all elements in $U$ that are not in $Q$ . These are the odd numbers in U.
$Q^{'} = {1, 3, 5, 7, 9, 11, 13, 15}$ .
Finally, find the union of P and $Q^{'}$ , denoted $P \cup Q^{'}$ . This set contains all elements from P or $Q^{'}$ , without duplication.
$P = {2, 3, 5, 7, 11, 13}$ .
$Q^{'} = {1, 3, 5, 7, 9, 11, 13, 15}$ .
$P \cup Q^{'} = {1, 2, 3, 5, 7, 9, 11, 13, 15}$ .
Therefore, the correct set is ${1, 2, 3, 5, 7, 9, 11, 13, 15}$ .

Answer: A

Q4 Hard

In a high school, a survey was conducted on student preferences for three subjects: History (H), Art (A), and Music (M). The results are given in percentages:
- 70% of students like History.
- 60% of students like Art.
- 50% of students like Music.
- 40% of students like History and Art.
- 35% of students like Art and Music.
- 30% of students like History and Music.
If 15% of students like none of the three subjects, what percentage of students like all three subjects?

A 5%

B 10%

C 15%

D 20%

Show Solution

Let H, A, and M represent the sets of students who like History, Art, and Music, respectively.

We are given the following percentages:
$∣ H ∣ = 70%$
$∣ A ∣ = 60%$
$∣ M ∣ = 50%$
$∣ H \cap A ∣ = 40%$
$∣ A \cap M ∣ = 35%$
$∣ H \cap M ∣ = 30%$
$∣ (H \cup A \cup M)^{'} ∣ = 15%$ (students who like none of the subjects)
From the percentage of students who like none, we can find the percentage of students who like at least one subject:
$∣ H \cup A \cup M ∣ = 100% - ∣ (H \cup A \cup M)^{'} ∣ = 100% - 15% = 85%$ .
Now, we use the Principle of Inclusion-Exclusion for three sets:
$∣ H \cup A \cup M ∣ = ∣ H ∣ + ∣ A ∣ + ∣ M ∣ - (∣ H \cap A ∣ + ∣ A \cap M ∣ + ∣ H \cap M ∣) + ∣ H \cap A \cap M ∣$
Substitute the given values into the formula:
$85% = 70% + 60% + 50% - (40% + 35% + 30%) + ∣ H \cap A \cap M ∣$
Calculate the sum of individual percentages:
$70% + 60% + 50% = 180%$
Calculate the sum of percentages for pairs of subjects:
$40% + 35% + 30% = 105%$
Substitute these sums back into the formula:
$85% = 180% - 105% + ∣ H \cap A \cap M ∣$
$85% = 75% + ∣ H \cap A \cap M ∣$
Solve for $∣ H \cap A \cap M ∣$ :
$∣ H \cap A \cap M ∣ = 85% - 75%$
$∣ H \cap A \cap M ∣ = 10%$
Therefore, 10% of students like all three subjects.

Answer: B

Tips & Strategies

Draw it out! If an ISEE question gives you a word problem about two groups but no picture, draw your own Venn diagram. Two overlapping circles can make a confusing problem super easy to see.
Work from the inside out. When filling in a Venn diagram, always write the number for the "Both" (intersection) section first, then figure out the rest!

Common Mistakes

Watch out for the "Double Count" trap! If a problem says "15 kids play soccer and 10 play tennis," you can't just add $15 + 10 = 25$ to find the total. Some of those soccer players might also play tennis. Always look for the "Both" group and subtract it!

Frequently Asked Questions

What does the word "Complement" mean in Set Theory?

Think of it as the "opposite." The complement of a set is everything that is NOT in that set. If Set A is "all the dogs," the complement is every animal that is not a dog!

Will the ISEE ask me about more than two overlapping circles?

For the Lower and Middle Level ISEE, you will usually only see two circles. The Upper Level might occasionally sneak in three, but the same rules apply—just work from the very center overlap outwards!

What if I completely blank on a Venn diagram question during the test?

Guess! The ISEE has absolutely no penalty for wrong answers. If you are stuck, eliminate any answers that seem way too big or way too small, pick your favorite letter, and move on.

Generate Unlimited Practice Questions

Learn This Topic

Practice Questions

Tips & Strategies

Common Mistakes

Frequently Asked Questions

Related Topics

More Data, Statistics & Probability

Other Levels

Browse

Generate Unlimited Practice Questions