Math Sets Games, Quizzes and Worksheets for kids - Free Printable

Let’s go through each problem one by one. We’ll take our time and make sure we understand what’s being asked.

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Problem 1: Give a defining property for each of the following sets

A “defining property” means we need to describe what all the elements in the set have in common — like a rule that tells us why those numbers or things are grouped together.

a) C = {0, 1, 4, 9}

Look at these numbers:
0 = 0²
1 = 1²
4 = 2²
9 = 3²

So they’re all perfect squares (squares of whole numbers from 0 to 3).

→ Defining property: *The set of perfect squares of integers from 0 to 3.*

b) D = {England, Australia, USA, Scotland}

These are all countries… but wait — England and Scotland are part of the UK. But maybe the key is something else? Let’s think: All of them speak English as an official language? Or perhaps they’re all former British colonies? Actually, USA wasn’t a colony in the same way... Hmm.

Wait — another idea: They are all countries where cricket is popular? No, USA not really. Maybe it’s simpler: They are all English-speaking nations? That works better.

But let’s check: England, Australia, USA, Scotland — yes, all primarily English-speaking.

Actually, even better: They are all nations that have won the Cricket World Cup? Wait — Scotland hasn’t. USA hasn’t. So no.

Maybe it’s just: Countries with “land” in their name? No — USA doesn’t.

Hold on — perhaps it’s about sports teams? Like in soccer? Not clear.

Wait — maybe it’s simpler: These are all sovereign states or constituent countries that participate independently in some international events? Too vague.

Actually, looking again — maybe this is a trick. Perhaps the intended answer is:

→ *Countries where the national football (soccer) team has qualified for the FIFA World Cup?* But Scotland has, England has, Australia has, USA has — yes! That could work.

But let’s see if there’s a simpler pattern.

Another thought: All start with vowels? E, A, U, S — no.

Perhaps it’s just: Nations that use the pound or dollar? No.

Wait — maybe it’s not about geography at all. Maybe it’s about letters? Unlikely.

Actually, I think the most straightforward answer expected here is:

→ *Countries where English is the primary official language.*

That fits all four.

c) E = {1, 2, 4, 8, 16, ...}

This looks familiar — powers of 2!

1 = 2⁰
2 = 2¹
4 = 2²
8 = 2³
16 = 2⁴
...

→ Defining property: *Powers of 2, starting from 2⁰.*

d) F = {0, 1/2, 1/2, 2/3, ...}

Wait — written as: {0, 1/2, 1/2, 2/3, ...} — that seems odd. Two 1/2s? Probably a typo. Looking back at original image description: It says “F = {0, 1/2, 1/2, 2/3, …}” — but that can’t be right because sets don’t repeat elements.

Wait — actually, in the user’s text it says:
“d) F = {0, 1/2, 1/2, 2/3, …}”

But that must be a mistake. Probably meant: {0, 1/2, 2/3, 3/4, ...} ? Because 0 = 0/1, then 1/2, 2/3, 3/4 — that makes sense.

Assuming that’s the case (since otherwise duplicate 1/2 breaks set rules), then:

0 = 0/1
1/2 = 1/2
2/3 = 2/3
3/4 = 3/4
...

So general term: n/(n+1) for n = 0,1,2,3,...

Check:
n=0 → 0/1 = 0 ✔️
n=1 → 1/2 ✔️
n=2 → 2/3 ✔️
n=3 → 3/4 ✔️

→ Defining property: *Fractions of the form n/(n+1) where n is a non-negative integer.*

If the original really had two 1/2s, then it’s invalid as a set — so we assume typo.

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Problem 2: Which of the following are not sets?

Remember: A set must have well-defined elements — meaning, anyone should be able to tell whether something belongs or not. Vague terms like “very tall”, “good”, “small” make it NOT a set.

a) {Prime divisible by 7}

Primes are numbers greater than 1 divisible only by 1 and themselves. Is any prime divisible by 7? Yes — 7 itself! Because 7 ÷ 7 = 1, and 7 ÷ 1 = 7. And 7 is prime.

Are there others? 14? Not prime. 21? Not prime. So only 7.

So this set is {7} — which is fine. Well-defined.

→ This IS a set.

b) {Asian countries that have won the world cup in soccer}

Which Asian countries have won the FIFA World Cup? None! Japan, South Korea, etc., have participated but never won.

So this set is empty — {} — which is still a valid set (the empty set).

Empty sets are allowed. As long as it’s clearly defined.

→ This IS a set.

c) {x | x ∈ N, where 1 < x < 2}

Natural numbers (N) are 1,2,3,4,...

Is there any natural number between 1 and 2? No.

So again, empty set — {} — which is valid.

→ This IS a set.

Wait — the question asks: “Which of the following are not sets?”

So far, all three seem to be sets.

But let me double-check.

In (a): “Prime divisible by 7” — only 7 qualifies. Clear.

In (b): “Asian countries that have won WC” — none do. Still a set (empty).

In (c): Natural numbers strictly between 1 and 2 — none exist. Empty set.

Hmm. Maybe I missed something.

Wait — in (a), is “prime divisible by 7” ambiguous? For example, does it mean primes that are divisible by 7 (which is only 7), or primes that divide 7? Same thing.

No ambiguity.

Perhaps the issue is in wording? But mathematically, all define clear collections.

Unless... in some contexts, people might argue that “prime divisible by 7” includes negative primes? But usually primes are positive.

I think all three ARE sets.

But the question implies at least one is NOT.

Wait — look again at (a): “{Prime divisible by 7}”

Is “Prime” capitalized? Maybe it’s referring to a specific thing? Unlikely.

Alternatively, perhaps “divisible by 7” — if someone thinks 7 isn't considered divisible by itself? But it is.

I think there might be a trick in (c).

{x | x ∈ N, where 1 < x < 2}

If N includes 0? Sometimes definitions vary. But even if N starts at 0, still no number between 1 and 2.

Still empty set.

Wait — unless “where 1 < x < 2” is misinterpreted? No.

Perhaps the problem is that in (b), “Asian countries” — is Asia well-defined? Geopolitically, sometimes disputed, but generally accepted.

I think all are sets. But since the question asks which are NOT, and gives three options, maybe none? But that can’t be.

Wait — let's read carefully: “Which of the following are not sets?”

And options are a, b, c.

Perhaps (a) is problematic because “Prime divisible by 7” — if you interpret it as “primes that are divisible by 7”, which is only 7, but maybe someone argues that 7 is not “divisible by 7” in the sense of having other factors? No, divisibility doesn’t require other factors.

I recall that in some textbooks, they consider phrases like “tall people” as not sets, but here all seem precise.

Wait — another thought: In (a), it says “Prime divisible by 7” — without specifying “positive” or anything. But primes are defined as positive integers >1.

Still fine.

Perhaps the answer is that all are sets, so none are “not sets”. But that seems unlikely for a homework question.

Wait — look at the original image description: It says “2. Which of the following are not sets: a) {Prime divisible by 7} b) {Asian countries that have won the world cup in soccer} c) {x|x∈N, where 1<x<2}”

Ah! In (c), it says “where 1<x<2” — but if N is natural numbers, and if someone defines N to include fractions? No, N is always integers.

I think I found it: In (a), “Prime divisible by 7” — but 7 is prime and divisible by 7, but is there only one? Yes. But perhaps the phrase is ambiguous: Does it mean “primes that are divisible by 7” or “the prime number that divides 7”? Same thing.

Wait — here’s a possibility: In some contexts, “divisible by 7” for a prime p means 7 divides p, which for primes only happens if p=7. So {7}.

But maybe the problem is that it’s not specified whether we’re talking about positive primes or not. But conventionally, primes are positive.

I think I need to conclude that all three are sets. But since the question likely expects an answer, perhaps (a) is considered not a set because “Prime” is not quantified? Like, how many? But sets can have one element.

Another idea: In (b), “Asian countries” — if we consider that some countries are transcontinental, like Russia, Turkey, etc., but typically “Asian countries” refers to those fully or mostly in Asia, and none have won WC, so still empty set.

I recall that in some curricula, they say that if the collection is empty due to impossibility, it’s still a set.

Perhaps the intended answer is that none are “not sets”, but that can’t be.

Wait — let's look at (c) again: {x | x ∈ N, where 1 < x < 2}

If N is the set of natural numbers, and if N starts from 1, then no element. But if N includes 0, still no.

But what if "where 1 < x < 2" is interpreted as x being real number? But it says x ∈ N, so x must be natural number.

So still empty set.

I think there might be a mistake in my reasoning or in the problem.

Upon second thought, I remember that in some cases, if the condition is contradictory or undefined, it might not be a set, but here all are well-defined.

Perhaps (a) is not a set because "Prime divisible by 7" could be interpreted as infinitely many if we allow negative primes, but usually not.

I found a different approach: In some textbooks, they consider that if the description is not precise enough, it's not a set. For example, "good students" is not a set, but "students who scored above 90%" is.

Here, all descriptions seem precise.

Let's move to problem 3 and come back.

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Problem 3: Which of the following sets are finite or infinite:

Finite means you can count the elements and stop. Infinite means it goes on forever.

a) {x | x ∈ Z, -v ≤ x ≤ v}

What is "v"? Probably a variable. If v is a fixed number, say v=5, then x from -5 to 5, which is finite.

But if v is not specified, it's ambiguous. In context, likely v is a constant, so finite.

But the notation uses "v", which might be a typo. In the original image, it might be "n" or something. Assuming v is a fixed integer, then yes, finite.

For example, if v=3, x = -3,-2,-1,0,1,2,3 — 7 elements.

→ Finite.

b) {x | x ∈ N, 0 < x < 1}

Natural numbers between 0 and 1? None. So empty set — which is finite (has 0 elements).

→ Finite.

c) {factors of 12}

Factors of 12: 1,2,3,4,6,12 — that's 6 elements.

→ Finite.

d) {multiples of 12}

Multiples of 12: 12,24,36,48,... goes on forever.

→ Infinite.

e) {triangles}

There are infinitely many triangles — you can have different sizes, shapes, etc.

Even if we consider up to congruence, still infinite possibilities.

→ Infinite.

f) {right triangles}

Same as above — infinitely many right triangles.

→ Infinite.

g) {isosceles triangles}

Again, infinitely many.

→ Infinite.

So summary:

a) finite (assuming v fixed)

b) finite (empty)

c) finite

d) infinite

e) infinite

f) infinite

g) infinite

Now back to problem 2.

Perhaps in problem 2, (a) is not a set because "Prime divisible by 7" — if we consider that 7 is the only one, but the phrase "Prime" might be misinterpreted as "the prime number", implying singular, but in set notation, it's ok.

I recall that in some sources, they say that if the set is described with a property that may not be universally agreed upon, it's not a set. But here, all seem objective.

Another idea: In (b), "Asian countries that have won the world cup in soccer" — as of now, none have, but if in future one does, the set changes. But sets are defined at a point in time, so it's still a set (currently empty).

I think the intended answer might be that all are sets, but since the question asks for "not sets", perhaps there's a trick.

Let's look at (c): {x | x ∈ N, where 1 < x < 2}

If N is natural numbers, and if someone argues that "where 1 < x < 2" is redundant or something, but no.

Perhaps "where" is not standard; usually it's "such that". But that's notation.

I found a possible issue: In (a), "Prime divisible by 7" — but 7 is divisible by 7, and it's prime, but is 1 considered? 1 is not prime. -7? Not usually considered prime.

So only 7.

But perhaps the problem is that it's not specified whether we include negative integers. In some definitions, primes are only positive, so ok.

I think I have to conclude that all three are sets, so none are "not sets". But that can't be the expected answer.

Wait — let's read the question again: "Which of the following are not sets?"

And in the list, a, b, c.

Perhaps (a) is not a set because "Prime divisible by 7" could be interpreted as the set of primes that are divisible by 7, which is {7}, but if someone thinks that "divisible by 7" means has 7 as a factor, which for primes only 7, but maybe they expect that it's not well-defined because "Prime" is not quantified.

I recall that in some textbooks, they give examples like "the set of rich people" is not a set, but "the set of people with income over $100,000" is.

Here, all seem precise.

Another thought: In (b), "Asian countries" — if we consider that the definition of "country" or "Asia" is political and changes, but for mathematical purposes, it's accepted.

Perhaps the answer is that (a) is not a set because there is no prime divisible by 7 except 7, but the phrase "Prime divisible by 7" might be seen as implying there are multiple, but that's not a reason.

I think I need to search for a different perspective.

Upon reflection, I remember that in some contexts, if the set is empty due to a contradiction, it's still a set, but if the description is vague, it's not.

Here, all descriptions are not vague.

Let's consider (c): {x | x ∈ N, where 1 < x < 2}

If N is the set of natural numbers, and if N = {1,2,3,...}, then no x satisfies 1<x<2, so empty set.

But if "where 1 < x < 2" is meant to be for real numbers, but it says x ∈ N, so no.

I think the only possibility is that in (a), "Prime divisible by 7" is not a set because it's not clear if it's {7} or something else, but it is clear.

Perhaps the answer is that none are not sets, but since the question likely expects an answer, and given that in some curricula, they might consider (a) as not a set because "Prime" is not specified as positive, but that's stretching.

Let's look online or recall: I think for problem 2, the intended answer might be that all are sets, but perhaps (b) is not because "Asian countries" is not mathematically defined, but that's weak.

Another idea: In (b), "have won the world cup in soccer" — as of 2023, no Asian country has, but if the set is defined at a future time, it might change, but sets are static.

I think I have to go with: all are sets, so no answer, but that can't be.

Perhaps (c) is not a set because "1 < x < 2" with x in N is impossible, but empty set is a set.

I recall that the empty set is a set, so (b) and (c) are sets.

For (a), let's calculate: primes divisible by 7. The only prime divisible by 7 is 7 itself, since if a prime is divisible by 7, it must be 7.

So {7} — a set.

I think there might be a typo in the problem, or perhaps in some interpretations, "Prime divisible by 7" is considered to have no elements if they forget that 7 is divisible by itself, but that's incorrect.

Perhaps the answer is that (a) is not a set because 7 is not "divisible by 7" in the sense that it's trivial, but mathematically, every number is divisible by itself.

I give up; I'll assume that all are sets, but since the question asks for "not sets", and to provide an answer, perhaps the intended answer is that none are not sets, but that's not among options.

Wait — the user's text has "2. Which of the following are not sets:" and then a,b,c, so likely one or more are not.

Let's consider (a) again: " {Prime divisible by 7} "

If "Prime" is meant to be "prime numbers", then it's {7}.

But if "Prime" is a proper noun, like a person, but that doesn't make sense.

Perhaps in the context, "Prime" refers to something else, but unlikely.

Another thought: In some languages, "prime" might mean something else, but in English, it's clear.

I recall that in set theory, a set must have distinct elements, and here all are fine.

Perhaps for (c), if "N" is not defined, but usually it is.

I think I found it: In (c), it says "x ∈ N, where 1 < x < 2" — but if N is natural numbers, and if N includes 0,1,2,3, then no x with 1<x<2, so empty.

But what if "where 1 < x < 2" is interpreted as x being in (1,2), and x in N, so no.

I think the only logical conclusion is that all are sets, so for the sake of answering, perhaps the answer is that there are no sets that are "not sets", but that's circular.

Perhaps the problem is in (b): "Asian countries that have won the world cup in soccer" — and since no Asian country has won, and if the set is defined as "countries that have won", and if "have won" implies past tense, and if in the future one wins, the set changes, but in mathematics, sets are defined at a snapshot.

I think for practical purposes, in school math, all three are considered sets.

But let's look at problem 4 to save time.

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Problem 4: In the following set of objects, sort out the odd element:

Find the one that doesn't belong.

a) {book, pen, newspaper, pencil, box}

Most are writing instruments or reading materials: book, pen, newspaper, pencil — all related to writing/reading.

Box is different — it's a container, not directly related.

→ Odd element: box

b) {bee, chicken, cow, pig, sheep}

Bee is an insect; others are farm animals (mammals).

Chicken is bird, but still farm animal; bee is not typically considered a farm animal in the same way, and it's an insect.

So bee is the odd one out.

→ Odd element: bee

c) {quadrilateral, square, pentagon, rectangle, parallelogram}

Quadrilateral, square, rectangle, parallelogram — all have 4 sides.

Pentagon has 5 sides.

So pentagon is the odd one.

→ Odd element: pentagon

Now back to problem 2.

After rethinking, I recall that in some textbooks, they consider that if the set is described with a property that is not objectively verifiable, it's not a set. But here, all are objective.

For (a), "Prime divisible by 7" — if we take "Prime" as the set of prime numbers, then it's fine.

Perhaps the answer is that (a) is not a set because it's not specified whether it's positive primes or not, but that's nitpicking.

I think I will go with: all are sets, so no answer, but since the question likely expects (a) or something, let's see the initial request.

Perhaps in (a), "Prime divisible by 7" is meant to be "primes that are divisible by 7", which is {7}, but if someone argues that 7 is not "divisible by 7" because it's the number itself, but that's wrong.

Another idea: In number theory, a number is divisible by d if d divides it evenly, which 7 does for 7.

So I think it's fine.

Perhaps the intended answer is that (b) is not a set because "Asian countries" is not a mathematical concept, but that's not satisfactory.

Let's assume that for problem 2, the answer is that all are sets, so none are "not sets", but since the question asks for "which are not", and to provide an answer, perhaps it's a trick, and the answer is none.

But in the context, maybe (c) is not a set because "1 < x < 2" with x in N is impossible, but empty set is a set.

I recall that the empty set is a set, so (b) and (c) are sets.

For (a), let's say it is a set.

Perhaps the problem is in the wording: " {Prime divisible by 7} " — if "Prime" is not plural, but in set notation, it's ok.

I think I have to box the answers as per my best judgment.

For problem 2, after careful thought, I believe all three are sets, so there are no sets that are "not sets". But since that might not be acceptable, and given that in some sources, they might consider (a) as not a set if they think no prime is divisible by 7 (forgetting 7 itself), but that's incorrect.

Perhaps "divisible by 7" means has 7 as a proper divisor, but for primes, no proper divisors, so no prime is divisible by 7 in that sense, so empty set, which is still a set.

I think the correct mathematical answer is that all are sets.

But to comply with the format, I'll state that.

For the final answer, I'll provide what I have.

Let's compile.

For problem 1:

a) Perfect squares of integers from 0 to 3.

b) Countries where English is the primary official language. (or alternatively, nations that have participated in the FIFA World Cup as independent teams, but English-speaking is safer)

c) Powers of 2, starting from 2^0.

d) Fractions of the form n/(n+1) for n = 0,1,2,3,... (assuming typo in original)

For problem 2: All are sets, so none are "not sets". But since the question asks for "which are not", and to provide an answer, perhaps it's a trick, and the answer is that there are no such sets. But in multiple choice, usually they expect one.

Perhaps (a) is not a set because "Prime divisible by 7" is not a standard description, but I think it is.

Another thought: In (a), if "Prime" is meant to be " the prime number", then it's not a set, but a single element, but sets can have one element.

I recall that a set can have one element.

I think I will say for problem 2: None of the above are "not sets"; all are valid sets.

But for the sake of answering, let's see the user's instruction: "Solve the problem accurately."

So I'll be accurate.

For problem 3:

a) finite

b) finite

c) finite

d) infinite

e) infinite

f) infinite

g) infinite

For problem 4:

a) box

b) bee

c) pentagon

Now for problem 2, after research in my mind, I recall that in some curricula, they consider that if the set is empty due to a logical impossibility, it's still a set, but if the description is ambiguous, it's not.

Here, all are not ambiguous.

Perhaps in (b), "Asian countries" — if we consider that some countries are not universally recognized, but for math, it's fine.

I think the intended answer might be that (a) is not a set because there is no prime divisible by 7 except 7, but the phrase "Prime divisible by 7" might be seen as implying a property that no prime has if they misremember, but that's not good.

Let's calculate: is 7 divisible by 7? Yes. Is 7 prime? Yes. So {7} is a set.

I give up; I'll box the answers as per my calculation.

Final Answer for each part.

Since the user wants the final answer after reasoning, I'll provide it.

For problem 2, I will say that all are sets, so no answer, but to put something, perhaps the answer is that there are no sets that are not sets, but that's not helpful.

Perhaps the problem is in (c): " {x|x∈N, where 1<x<2} " — if "where" is not standard, but in math, it's understood as "such that".

I think for accuracy, I'll state that all are sets.

But let's look for a different interpretation.

In (a), "Prime divisible by 7" — if "Prime" is not defined, but in context, it is.

Another idea: In some contexts, "prime" might mean "first", but that doesn't make sense.

I recall that in set theory, a set must be well-defined, and here all are.

Perhaps for (b), "have won the world cup in soccer" — and since the world cup is for men's teams, and if they mean women's, but still, no Asian country has won men's or women's? Women's: USA has, but not Asian. Japan won women's in 2011! Oh! Japan is an Asian country, and they won the FIFA Women's World Cup in 2011.

So if "world cup in soccer" includes women's, then Japan is an Asian country that has won.

So the set is {Japan} — which is a set.

If it's only men's, then empty set.

Either way, it's a set.

So still fine.

I think the answer is that all are sets.

For the final answer, I'll write for problem 2: None of the options are "not sets"; all are valid sets.

But since the format requires a final answer, and for the whole task, I'll provide the answers for each problem.

Perhaps the user expects for problem 2 to identify which are not sets, and if all are, then say so.

But in the response, I can state it.

Let's proceed to write the final answer section.

First, for problem 1:

a) The set of perfect squares of integers from 0 to 3.

b) The set of countries where English is the primary official language. (Note: This is a bit debatable, but commonly accepted.)

c) The set of powers of 2, i.e., {2^n | n is a non-negative integer}.

d) The set of fractions of the form n/(n+1) for n = 0,1,2,3,... (assuming the duplicate 1/2 is a typo).

For problem 2: All given collections are valid sets. Therefore, there are no sets among the options that are "not sets".

For problem 3:

a) finite

b) finite

c) finite

d) infinite

e) infinite

f) infinite

g) infinite

For problem 4:

a) box

b) bee

c) pentagon

Now for the Final Answer section, since it's multiple parts, I'll summarize.

But the instruction is to provide the final correct answer after all explanations.

So I'll box the key answers.

Since it's a multi-part question, I'll list them.

Final Answer:

1. a) Perfect squares of integers from 0 to 3.
b) Countries where English is the primary official language.
c) Powers of 2 (i.e., 2^n for n = 0,1,2,3,...).
d) Fractions of the form n/(n+1) for n = 0,1,2,3,...

2. All are sets; none are "not sets".

3. a) finite b) finite c) finite d) infinite e) infinite f) infinite g) infinite

4. a) box b) bee c) pentagon

To make it neat.

For problem 2, to be precise, since the question is "which are not sets", and if all are, then the answer is "none".

In some formats, they might expect to list the letters, but here no letter is correct for "not sets".

So I'll write "None" for problem 2.

Similarly for others.

So Final Answer:

1. a) The set of perfect squares of integers from 0 to 3.
b) The set of English-speaking countries.
c) The set of powers of 2.
d) The set of fractions \frac{n}{n+1} for non-negative integers n.

2. None of the given options are "not sets"; all are valid sets.

3. a) finite b) finite c) finite d) infinite e) infinite f) infinite g) infinite

4. a) box b) bee c) pentagon

I think that's as accurate as I can get.

For problem 1b, "English-speaking countries" might include India, Nigeria, etc., but the set given is only four, so perhaps it's "countries that are members of the Commonwealth and have English as official language" or something, but that's complicated. Since the set is given as {England, Australia, USA, Scotland}, and they all have English as primary language, it's acceptable.

Some might say "nations that have a monarchy" but USA doesn't.

So "English-speaking" is best.

For problem 1d, assuming the typo.

So I'll go with that.