What Makes a Valid Set?
Sets are the foundation of almost every topic you will meet in this course. Before you can combine them, compare them, or find what they share, you need to be able to answer one basic question: is this collection of things actually allowed to be a set in mathematics?
The answer does not require any calculation at all. It only requires you to read a description carefully and ask yourself one simple question. This lesson will teach you exactly what that question is, show you how to apply it quickly, and walk you through the kinds of descriptions that appear on the CSCA exam — including the ones designed to trick you.
You can learn anything. Let's do this!
The one rule every set must follow
A set is a collection of things. Those things can be numbers, people, countries, colours — anything at all. Each individual item inside a set is called an element (also called a member).
But not every collection of things qualifies as a set in mathematics. For a collection to be a proper mathematical set, it must be well-defined. This is the single most important idea in this entire lesson.
Think of it like an official school register. If a teacher says "this class contains all students enrolled in Room 3B on 1st September 2024", then anyone can check the school's official records and know for certain who is in the class. That description is well-defined. But if the teacher says "this class contains all the smart students", two different teachers would give two completely different lists. That is not well-defined — and so it cannot be a set.
The test you should run for every description is this: Could two people read this and disagree about whether a specific object belongs? If yes, it is not well-defined. If no — if everyone would always agree — then it is well-defined, and it can be a set.
The correct answer is B. "Good" is a subjective judgment — one teacher might consider a student good because of grades, another because of behaviour, another because of effort. There is no single, universally agreed test for goodness. Because different observers would produce different lists, the collection is not well-defined and cannot be a set.
Option A is wrong because the size of a collection has nothing to do with whether it can be a set. A set can have a billion elements — what matters is whether membership is objectively clear. Option C is wrong because specifying the school more precisely would not fix the core problem, which is the word "good". Option D raises a real issue with time, but that is not the fundamental problem here — the description fails the well-defined test regardless of time.
Valid sets vs invalid sets — seeing the difference
Now that you understand the rule, let's look at real examples side by side. Study each one carefully. The goal is to train yourself to spot the difference instantly.
| Description | Valid? | Why |
|---|---|---|
| All even numbers between 1 and 10 | ✓ Yes | You can list them exactly: {2, 4, 6, 8, 10}. No disagreement possible. |
| All Year 10 students at a specific school during a specific semester | ✓ Yes | The school has an official register. You can check any name against it. |
| All countries in the United Nations | ✓ Yes | An official membership list exists. Membership is objectively checkable. |
| All people taller than 180 cm | ✓ Yes | 180 cm is a fixed, measurable boundary. You can measure anyone and know. |
| All real numbers very close to 1 | ✗ No | "Very close" has no fixed boundary. Is 0.99 close enough? What about 0.5? Nobody agrees. |
| All tall people in the world | ✗ No | "Tall" has no fixed cutoff. One person says 180 cm is tall; another says 175 cm. |
| Year 10 pupils with relatively good eyesight | ✗ No | "Relatively good" is a comparison with no fixed reference. Different doctors would disagree. |
| Highly talented teachers in Year 10 | ✗ No | "Highly talented" is a value judgment. Two principals would likely give different lists. |
Look at the pattern in the invalid examples. They all use language that depends on someone's personal opinion or a boundary that has not been precisely stated. The valid examples all share one quality: you could write a computer program to check whether any given object belongs, and that program would never need to make a judgment call.
Notice something important from the table: "all tall people" is invalid, but "all people taller than 180 cm" is valid. Adding a specific, measurable number to a vague description is often all it takes to turn an invalid description into a valid one. The word "tall" is the problem — not the concept of height.
The correct answer is D. "All integers x such that x² = 9" is a precise mathematical condition. You can test any integer: does squaring it give 9? The answer is always definitively yes or no. The set contains exactly {−3, 3} and there is no room for disagreement.
Option A fails because "hardworking" is a subjective judgment — no two teachers would necessarily agree on the same list. Option B fails because "popular" is a matter of opinion, and even if you use chart position, you would still need to specify which chart, which country, and which date. Option C fails because "approximately round" has no precise boundary — where does "approximately round" end and "not round enough" begin?
Warning words — the fastest way to spot an invalid set
On the CSCA exam, questions about valid sets often use descriptions that sound specific but are actually vague. The fastest way to catch these is to look for certain words. We call these warning words.
Whenever you see any of the following in a set description, treat it as a red flag:
| Warning word or phrase | Why it makes the set invalid |
|---|---|
| "Very close to", "approximately", "near", "around" | Suggests a vague region with no exact boundary. How close is "very close"? No one can agree. |
| "Relatively good/bad", "fairly", "quite", "somewhat" | Comparison words with no fixed reference point. Good compared to what? |
| "Talented", "smart", "clever", "gifted" | Describes ability, which is judged differently by different people. No universal test exists. |
| "Kind", "nice", "popular", "friendly" | Personality descriptions. One person's "kind" is another person's "just okay". Purely subjective. |
| "Hardworking", "successful", "important", "significant" | Value judgments that depend entirely on who is judging and what standard they use. |
On the flip side, these types of descriptions almost always produce valid sets:
The correct answer is A. The phrase "significantly tall" contains two warning words layered on top of each other: "significantly" and "tall". Both are vague. Even among NBA players who are all very tall, there is no agreed boundary for what counts as "significantly tall". One coach might say 210 cm, another might say 215 cm.
Option B is valid: "prime numbers less than 20" is a precise mathematical condition that produces the definite set {2, 3, 5, 7, 11, 13, 17, 19}. Option C is valid: the university has official enrollment records for a specific semester — membership is checkable. Option D is valid: x² − 4 = 0 has exactly two solutions, −2 and 2, and any number either satisfies this equation or it does not.
A real exam question, fully worked
Let's apply everything to the exact type of question that appears on the CSCA exam.
Which of the following objects can form a set?
A. All real numbers very close to 1
B. All Year 10 students at Peking University Affiliated Middle School Yunnan Experimental School during the second semester of the 2020–2021 academic year
C. Year 10 pupils with relatively good eyesight
D. Highly talented teachers in Year 10
Before even thinking about the options, run the well-defined test on each one.
Option A — "very close to 1". This is a warning phrase. Is 0.99 close enough? Is 0.5? Is 0.001? There is no agreed boundary, so two people could easily produce different lists. Not valid.
Option B — a specific school, a specific year group, a specific semester. The school has an official register for that semester. Anyone can check any student's name against that record. There is no room for disagreement. Valid.
Option C — "relatively good eyesight". The word "relatively" is a warning word — it signals a comparison with no fixed reference point. Good compared to what? Different doctors applying different standards would produce different lists. Not valid.
Option D — "highly talented teachers". The phrase "highly talented" is a value judgment. Two school principals would almost certainly disagree on who qualifies. Not valid.
The answer is B — and you can get there in about ten seconds once you know what to look for.
The correct answer is C. The problem word is "tall": it has no fixed boundary. "Taller than 175 cm" replaces it with a number anyone can measure. For every student in Class 2, a tape measure gives a clear yes or no, so everyone would make the same list.
Option A adds the warning word "very" but keeps "tall", so the boundary is still a matter of opinion. Option B changes the group (the whole school instead of Class 2), but the vague word "tall" is still there. A bigger group does not fix a vague rule. Option D makes membership depend on one person's opinion. A different teacher could make a different list, so it is still not well-defined.
One more trap: parameters that seem precise but aren't
Some exam questions are more subtle. They give you a description involving a parameter — a variable like a — and ask you to find what value of that parameter makes the set valid. The key insight is that a set's elements must be distinct — no element can appear twice.
Consider this situation: set A = {1, 2023, a²} and set B = {2023, a + 2}. The complement of B in A (written ∁AB) equals {1}. What must a be?
If ∁AB = {1}, that means {1} is the only element in A that is not in B. So both 2023 and a² must belong to B. Since 2023 is already in B, we need a² to equal the other element of B, which is a + 2. Solving a² = a + 2 gives us a = 2 or a = −1.
But we must check both answers against the distinctness rule: no element can appear twice in a set.
With a = 2: a² = 4 and a + 2 = 4, so B = {2023, 4}. Set A = {1, 2023, 4}. All elements are distinct, and ∁AB = {1}. This works. The answer is a = 2.
The lesson: whenever a question asks you to find a parameter for a set, always check your answer produces a valid set with no repeated elements.
The correct answer is C. Put each value into the three elements 2, a and a² − 2, then compare all three pairs. With a = 0: the elements are 2, 0 and 0² − 2 = −2. So A = {2, 0, −2}, and all three are different.
Each wrong option fails one pair, so you must check every pair, not just one. Option A: a = 2 is the same as the first element 2 (and a² − 2 = 2 as well), so A = {2, 2, 2}. Option B: a = −1 is different from 2, but a² − 2 = 1 − 2 = −1 equals a, so A = {2, −1, −1}. Option D: a = −2 is different from 2, but a² − 2 = 4 − 2 = 2 equals the first element, so A = {2, −2, 2}.
Your turn!
Practice questions
The correct answer is A. "All positive integers less than 100 that are divisible by 7" is a precise mathematical condition. You can test any positive integer: is it less than 100? Is it divisible by 7 with no remainder? The answers are always objectively yes or no. The set is {7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98}.
Option B fails because "beautiful" is a subjective judgment — two art critics would produce different lists. Option C fails because "fairly successful" uses the warning word "fairly" — no agreed test for success. Option D fails because "approximately average height" has a warning word ("approximately") and is vague — what counts as approximate?
The correct answer is D. This is a valid set because "x² − 5x + 6 = 0" is a precise mathematical condition. Any number either satisfies it or does not — there is no subjectivity and no grey area.
To find the elements, solve x² − 5x + 6 = 0. Factor: (x − 2)(x − 3) = 0, so x = 2 or x = 3. The set is {2, 3}. Check: 2² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓. And 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓.
Option A is wrong: an equation is one of the clearest ways to define a set, because every number either solves it or does not. Option B uses the wrong factor pair. 1 and 6 multiply to 6 but add to 7, so (x − 1)(x − 6) = x² − 7x + 6, which is a different equation. Option C is wrong twice. First, the equation does have real solutions: Δ = (−5)² − 4 × 1 × 6 = 25 − 24 = 1 > 0. Second, even an equation with no solutions gives a valid set — the empty set ∅, which contains nothing.
Things to remember
| Concept | What to remember |
|---|---|
| Valid set | Must be well-defined — membership can be decided objectively by anyone, without disagreement |
| Invalid set | Any description involving opinion, vagueness, or subjective judgment |
| Warning words | "Close to", "relatively", "talented", "popular", "good", "hardworking", "approximately", "significant" |
| Safe descriptions | Exact mathematical conditions, fixed numbers, specific dates, places, or official records |
| Quick fix | Adding an exact number to a vague description often makes it valid: "tall" → "taller than 180 cm" |
| Distinctness rule | Every element in a set must appear exactly once. When finding a parameter, always check your answer does not create a repeat. |
| Size doesn't matter | A set can be infinite — what matters is whether membership is objectively clear, not how many elements there are |