Set Theory
Set notation, Venn diagrams, union, intersection, and complement — excludes permutation/combination counting (see counting-combinations) and pure probability (see probability)
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Imagine you're throwing an epic pizza party! 🍕 You have a list of friends who love pepperoni and another list of friends who love extra cheese. Some lucky friends are on both lists because they love pepperoni AND extra cheese! In math, we call these lists "Sets."
A Set is just a group of things. It could be numbers, letters, or even your favorite video games. When we talk about sets on the SSAT, we usually use cool overlapping circles called Venn Diagrams. ⭕⭕
There are three magic words you need to know for Set Theory: 1. Intersection: This is where the circles overlap! It means "AND." Like friends who want pepperoni AND cheese. The math symbol looks like an upside-down U: . 2. Union: This means "OR." It's everyone in the first circle, everyone in the second circle, and everyone in the middle. It’s the whole party! The symbol looks like a big U: . 3. Complement: This is everything outside the circles. It’s the friends who don't want pizza at all (maybe they want tacos? 🌮).
On the SSAT, you'll often see word problems about kids playing different sports or joining clubs. Just draw your two circles, find the kids who do both (put them in the middle!), and you'll be a Set Theory superstar! 🌟
Practice Questions
5 practice questions for SSAT Upper Level
Sets A, B, and C are defined above. If is a number in both set A and set C, but not in set B, what is the value of ?
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- First, find the numbers that are in both set A and set C (the intersection of A and C). The numbers in set A are 2, 4, 6, 8, and 10. The numbers in set C are 2, 4, 12, and 14. The numbers that appear in both sets are 2 and 4.
Next, determine which of these numbers is NOT in set B. Set B contains 4, 8, 12, and 16. Since 4 is in set B, it is eliminated. The number 2 is not in set B. Therefore, .
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- Use the overlapping sets formula: Total = Group 1 + Group 2 - Both + Neither. Plugging in the given numbers: .
Combine the known values on the right side: .
The equation becomes . Solving for 'Both' gives 8, meaning 8 students play both soccer and basketball.
To find the number of students who play soccer but NOT basketball, subtract the students who play both from the total number of soccer players: .
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- Since every child likes at least one activity, the number of children who like neither is 0. Use the overlapping sets formula: Total = Swimming + Archery - Both.
Substitute the known values: .
This simplifies to .
Solving for the number of children who like both gives . Thus, 20 children like both activities.
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- If is in both Set X and Set Y, it must be a multiple of both 5 and 6. To find the common multiples of 5 and 6, first find their least common multiple (LCM). Since 5 and 6 share no common prime factors, their LCM is .
Therefore, must be a multiple of 30 (e.g., 30, 60, 90, 120...). Looking at the answer choices, only 90 is a multiple of 30 ().
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- First, find the number of tourists who speak both languages using the formula: Total = French + Spanish - Both + Neither.
Substitute the given values: .
Simplifying gives , so . There are 20 tourists who speak both languages.
To find the number who speak exactly one language, subtract the "Both" group from each individual language group.
French only: .
Spanish only: .
Add these together: tourists who speak exactly one language.
Tips & Strategies
- Always draw a Venn diagram! Overlapping circles make word problems so much easier to see. Start by filling in the middle part (the "Both" section) first.
- Watch out for the word "only." If a problem says "10 kids play ONLY soccer," that means you put 10 in the soccer circle but outside the overlap. If it just says "10 kids play soccer," some of those 10 might be in the overlap!
- Remember the symbols: looks like a big bowl holding everything (Union), and looks like a bridge connecting just the middle (Intersection).
Common Mistakes
- Watch out for double-counting the middle group! If 10 kids like apples, 10 like bananas, and 3 like both, the total isn't 20. You have to subtract the 3 so you don't count those kids twice!
- Don't forget the kids outside the circles. Sometimes a problem has a "Neither" group. Always check if the problem says "everyone does at least one activity" or if there are leftover kids.
Frequently Asked Questions
What does the weird upside-down U symbol mean?
That is the symbol for "Intersection" (). It means "AND." You are looking for the things that are in BOTH sets, right where the circles overlap!
Do I really need to draw the circles on the SSAT?
Yes! Drawing a quick Venn diagram in your test booklet takes just 5 seconds and helps prevent silly mistakes. Your brain loves pictures! 🧠
What if the SSAT gives me a problem with THREE circles?
Take a deep breath! Just start in the very center where all three circles overlap. Write that number down first, and then work your way outward step-by-step.
How is Set Theory different from Probability?
Set Theory is about counting how many items are in groups. Probability uses those numbers to find the chance of picking an item, which is usually written as a fraction like .