A Collection of Three Things Has Eight Different Subsets
Learn to write a set in roster and set-builder form, classify sets as empty, singleton, finite, equal or equivalent, count subsets with the two-to-the-n rule, and find the complement of a set in a universal set.
How many subsets does a set of three elements have?
Eight — and the reason is simple once you see it.
For each element you have exactly two choices: put it in the subset or leave it out. Three elements, each with two choices independently, give
That count includes the subset where you left everything out — the empty set — and the one where you took everything — the set itself. Both are genuine subsets, and forgetting them is why students often answer six.
So the number of subsets is for a set of elements. This page covers the first part of the ICSE Class 8 Mathematics chapter on sets: how to write them, how to classify them, how to count their subsets, and how to complement them.
For each element you have exactly two choices: put it in the subset or leave it out. Three elements, each with two choices independently, give
That count includes the subset where you left everything out — the empty set — and the one where you took everything — the set itself. Both are genuine subsets, and forgetting them is why students often answer six.
So the number of subsets is for a set of elements. This page covers the first part of the ICSE Class 8 Mathematics chapter on sets: how to write them, how to classify them, how to count their subsets, and how to complement them.
How do you write a set in roster form and in set-builder form?
A set is a well-defined collection of distinct objects. The objects are its elements or members, and they are written inside curly brackets.
Well-defined means there is no doubt whether a given thing belongs. The collection of *prime numbers less than is well defined; the collection of interesting numbers* is not, so it is not a set.
Roster form, also called tabular form, lists the elements separated by commas:
Set-builder form states the property the elements share:
The colon is read as such that, so the whole line reads *the set of all such that is a prime number less than ten*.
Converting between the two. The same set can always be written either way.
- becomes
- becomes
Two conventions that matter. Elements are not repeated — the letters of level form the set and not . And the order does not matter: and are the same set.
Membership notation. means * belongs to *, and means * does not belong to *.
When to use which form. Roster form is clearer for a small finite set. Set-builder form is essential for an infinite set, since you cannot list all of it — so the set of natural numbers must be written , or at best as with dots standing in for a property.
Well-defined means there is no doubt whether a given thing belongs. The collection of *prime numbers less than is well defined; the collection of interesting numbers* is not, so it is not a set.
Roster form, also called tabular form, lists the elements separated by commas:
Set-builder form states the property the elements share:
The colon is read as such that, so the whole line reads *the set of all such that is a prime number less than ten*.
Converting between the two. The same set can always be written either way.
- becomes
- becomes
Two conventions that matter. Elements are not repeated — the letters of level form the set and not . And the order does not matter: and are the same set.
Membership notation. means * belongs to *, and means * does not belong to *.
When to use which form. Roster form is clearer for a small finite set. Set-builder form is essential for an infinite set, since you cannot list all of it — so the set of natural numbers must be written , or at best as with dots standing in for a property.
What are the different types of sets?
Six classifications, and each answers a different question about the set.
Empty set — a set with no elements, written or . Also called the null or void set.
Singleton set — a set with exactly one element.
Finite set — the number of elements can be counted and comes to an end. is finite.
Infinite set — the elements never end. The set of natural numbers is infinite.
Cardinal number — written , it is the number of distinct elements in a finite set. For , ; and .
Equal sets — two sets are equal if they have exactly the same elements, written . Order and repetition are ignored, so
Equivalent sets — two sets are equivalent if they have the same cardinal number, whatever their elements are. So
are equivalent — both have three elements — but they are not equal, since not one element is shared.
The one-way relationship to remember. Equal sets are always equivalent, because identical elements means an identical count. But equivalent sets need not be equal, as the example above shows. Treating the two words as interchangeable is the commonest mistake in this section, and the question are these sets equal or equivalent? is set precisely to test it.
A boundary case worth knowing. and are not the same. The first is the empty set, with . The second is a singleton whose one element happens to be the empty set, so . Similarly is a singleton and not empty — it contains one element, which happens to be the number zero.
Empty set — a set with no elements, written or . Also called the null or void set.
Singleton set — a set with exactly one element.
Finite set — the number of elements can be counted and comes to an end. is finite.
Infinite set — the elements never end. The set of natural numbers is infinite.
Cardinal number — written , it is the number of distinct elements in a finite set. For , ; and .
Equal sets — two sets are equal if they have exactly the same elements, written . Order and repetition are ignored, so
Equivalent sets — two sets are equivalent if they have the same cardinal number, whatever their elements are. So
are equivalent — both have three elements — but they are not equal, since not one element is shared.
The one-way relationship to remember. Equal sets are always equivalent, because identical elements means an identical count. But equivalent sets need not be equal, as the example above shows. Treating the two words as interchangeable is the commonest mistake in this section, and the question are these sets equal or equivalent? is set precisely to test it.
A boundary case worth knowing. and are not the same. The first is the empty set, with . The second is a singleton whose one element happens to be the empty set, so . Similarly is a singleton and not empty — it contains one element, which happens to be the number zero.
Formula
How do you count and list all the subsets of a set?
Set is a subset of set , written , if **every element of is also in .
It is a proper subset**, written , if and — so at least one element of is missing from .
Two subsets every set has:
- The set itself, since every element of is certainly in .
- The empty set, since it has no element that could fail to be in .
For a set of elements:
The proper count is one less because the set itself is excluded.
Worked example 1 — listing them all. Find all the subsets of .
Work by size:
- With no elements:
- With one: , ,
- With two: , ,
- With three:
That is subsets, matching . Of these, seven are proper — all except itself.
Listing by size is worth doing in that order, because it guarantees none is missed and none repeated.
Worked example 2 — counting only. How many subsets does a set of elements have?
Worked example 3 — working backwards. A set has subsets. How many elements does it have?
This is an exponential equation of the kind met in the chapter on exponents — same base, so equate the indices.
Worked example 4 — the empty set. How many subsets does have?
Exactly one — the empty set itself. And it has proper subsets, since the only subset available is the set itself.
The distinction that costs marks. and are different. For :
- is true — is an element
- is true — is a subset
- is wrong, because is not a set
- is wrong, because the elements of are numbers, not sets
It is a proper subset**, written , if and — so at least one element of is missing from .
Two subsets every set has:
- The set itself, since every element of is certainly in .
- The empty set, since it has no element that could fail to be in .
For a set of elements:
The proper count is one less because the set itself is excluded.
Worked example 1 — listing them all. Find all the subsets of .
Work by size:
- With no elements:
- With one: , ,
- With two: , ,
- With three:
That is subsets, matching . Of these, seven are proper — all except itself.
Listing by size is worth doing in that order, because it guarantees none is missed and none repeated.
Worked example 2 — counting only. How many subsets does a set of elements have?
Worked example 3 — working backwards. A set has subsets. How many elements does it have?
This is an exponential equation of the kind met in the chapter on exponents — same base, so equate the indices.
Worked example 4 — the empty set. How many subsets does have?
Exactly one — the empty set itself. And it has proper subsets, since the only subset available is the set itself.
The distinction that costs marks. and are different. For :
- is true — is an element
- is true — is a subset
- is wrong, because is not a set
- is wrong, because the elements of are numbers, not sets
What is a universal set, and how do you find a complement?
The universal set, written or , is the set containing all the elements under consideration in a particular discussion. Every other set in that discussion is a subset of it.
Which set is universal depends entirely on the context. Discussing the digits, ; discussing the students of a class, is that class.
The complement of , written , is the set of all elements of the universal set that are not in :
Worked example. Let and .
Then
Checking the counts:
This relation always holds, and it is a useful check:
Four properties of complements:
- — complementing twice returns the original set.
- — nothing is outside the universal set.
- — everything is outside the empty set.
- and have no element in common, and together they make up the whole of .
Worked example 2 — with a described set. Let and let be the set of multiples of in .
And . Correct.
Why the universal set must be stated. A complement has no meaning on its own. The complement of is only because was the numbers to . Had been the numbers to , the complement would have contained ten more elements.
So a question asking for a complement must give you a universal set, and an answer that does not say which was used is incomplete.
Which set is universal depends entirely on the context. Discussing the digits, ; discussing the students of a class, is that class.
The complement of , written , is the set of all elements of the universal set that are not in :
Worked example. Let and .
Then
Checking the counts:
This relation always holds, and it is a useful check:
Four properties of complements:
- — complementing twice returns the original set.
- — nothing is outside the universal set.
- — everything is outside the empty set.
- and have no element in common, and together they make up the whole of .
Worked example 2 — with a described set. Let and let be the set of multiples of in .
And . Correct.
Why the universal set must be stated. A complement has no meaning on its own. The complement of is only because was the numbers to . Had been the numbers to , the complement would have contained ten more elements.
So a question asking for a complement must give you a universal set, and an answer that does not say which was used is incomplete.
Exam tip
Exam tip: count the empty set and the set itself as subsets
When listing subsets, **always include and the set itself**. Omitting them is the most frequent error here, and it makes the count disagree with — which is a free check you should run every time.
List subsets by size — none, then one element, then two, and so on. It guarantees nothing is missed or duplicated.
Remember **proper subsets number **, excluding the set itself.
Keep and apart: relates an element to a set, relates a set to a set. For , write and , never the reverse.
For equal or equivalent, say which and why: equal means the same elements, equivalent means the same number of elements. Equal sets are always equivalent; the reverse is not true.
Note that , and are three different sets, with , and .
A complement needs a stated universal set — say which you used.
Use to check a complement, and remember .
And in set-builder form, read the colon as such that and state the property fully, including any restriction such as .
List subsets by size — none, then one element, then two, and so on. It guarantees nothing is missed or duplicated.
Remember **proper subsets number **, excluding the set itself.
Keep and apart: relates an element to a set, relates a set to a set. For , write and , never the reverse.
For equal or equivalent, say which and why: equal means the same elements, equivalent means the same number of elements. Equal sets are always equivalent; the reverse is not true.
Note that , and are three different sets, with , and .
A complement needs a stated universal set — say which you used.
Use to check a complement, and remember .
And in set-builder form, read the colon as such that and state the property fully, including any restriction such as .
Did you know
Why does the empty set count as a subset of every set?
It sounds like a technicality invented to make a formula work. It is not — it follows from what subset means.
says: **every element of is also in . To show that this fails**, you would have to produce an element of that is missing from .
Now try to do that for the empty set. You would need to find an element of — and there are none to find. So the requirement can never be broken, and the statement holds for every set without exception.
The same reasoning is why the count works out to exactly . Choosing a subset means making an in-or-out decision for each of elements, and out for all of them is a perfectly ordinary set of decisions — it produces .
Which is a neat demonstration of how mathematical definitions behave. The empty set was not added to the list of subsets to tidy up the arithmetic; the arithmetic came out tidy because the definition already included it.
says: **every element of is also in . To show that this fails**, you would have to produce an element of that is missing from .
Now try to do that for the empty set. You would need to find an element of — and there are none to find. So the requirement can never be broken, and the statement holds for every set without exception.
The same reasoning is why the count works out to exactly . Choosing a subset means making an in-or-out decision for each of elements, and out for all of them is a perfectly ordinary set of decisions — it produces .
Which is a neat demonstration of how mathematical definitions behave. The empty set was not added to the list of subsets to tidy up the arithmetic; the arithmetic came out tidy because the definition already included it.
Key takeaways
Sets, subsets and complements: quick revision
- A set is a well-defined collection of distinct objects, written in curly brackets. Elements are never repeated and the order does not matter.
- Roster form lists the elements: . Set-builder form states the property: , with the colon read as such that.
- Use ** for membership and for non-membership.
- Empty set** has no elements; a singleton has exactly one; a finite set ends; an infinite set does not.
- Cardinal number counts distinct elements — .
- Equal sets have the same elements; equivalent sets have the same number of elements. Equal implies equivalent, but not the reverse.
- , and are three different sets, with , and .
- **** if every element of is in ; a proper subset also requires .
- Every set has itself and ** as subsets.
- Number of subsets ; proper subsets **. So has subsets and proper ones; a set of has and ; a set with subsets has , so ; and has subset.
- List subsets by size — for a three-element set.
- Keep ** (element to set) apart from (set to set).
- The universal set** holds everything under consideration; the complement .
- With and : , and .
- Properties: , , , and with share nothing and together fill .
- A complement is meaningless without a stated universal set.
Try listing every subset of a four-element set by size and checking the total against — the count either confirms your list or tells you exactly how many you missed.
- Roster form lists the elements: . Set-builder form states the property: , with the colon read as such that.
- Use ** for membership and for non-membership.
- Empty set** has no elements; a singleton has exactly one; a finite set ends; an infinite set does not.
- Cardinal number counts distinct elements — .
- Equal sets have the same elements; equivalent sets have the same number of elements. Equal implies equivalent, but not the reverse.
- , and are three different sets, with , and .
- **** if every element of is in ; a proper subset also requires .
- Every set has itself and ** as subsets.
- Number of subsets ; proper subsets **. So has subsets and proper ones; a set of has and ; a set with subsets has , so ; and has subset.
- List subsets by size — for a three-element set.
- Keep ** (element to set) apart from (set to set).
- The universal set** holds everything under consideration; the complement .
- With and : , and .
- Properties: , , , and with share nothing and together fill .
- A complement is meaningless without a stated universal set.
Try listing every subset of a four-element set by size and checking the total against — the count either confirms your list or tells you exactly how many you missed.