Class 8 · English Version · Mathematics
2026 Class 8 Mathematics · English Version · Chapter 7 · Chapter companion part
Describe well-defined sets, convert roster and set-builder forms, include every subset and take complements inside a stated universe.
Class 8 English Version: part A of the chapter 7 companion. Eight sections, 30 written tasks and a notebook challenge. Selected companion coverage, not complete textbook coverage; use all three parts. Counts are checked, while rosters, equivalent set-builder rules, diagrams and general proofs need guardian or teacher review.
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A set is a well-defined collection of distinct objects called elements. For A={2,4,6}, 4∈A means 4 belongs to A and 5∉A means it does not. Reordering or repeating entries does not change a set: {2,4,4,6}={6,2,4}. A rule such as integers from 1 through 5 determines membership; an unspecified collection of nice numbers does not. Elements need not be lowercase letters or even numbers.
The braces in {2} make a set whose one element is 2. For A={2,4}, write 2∈A but {2}⊆A. The symbol ⊆ means every element on its left is also on its right. The entire set A is a subset of itself. A roster, also called tabular form, lists the distinct elements in braces; commas separate elements, not digits inside one number.
Set-builder form names a variable and a condition, such as {x∈N: x is a multiple of 4 and 1≤x≤20}={4,8,12,16,20}. Here N={1,2,3,...}; some books include zero in N, so our convention is explicit. The colon means such that. If x is allowed to be any integer, even and x≤20 would also include zero and infinitely many negatives. A finite source list needs its intended lower bound stated.
A finite set has some non-negative whole number of elements. A huge finite set does not become infinite because counting it is inconvenient. The natural numbers continue without a last member and are infinite. They can be listed 1,2,3,... and are called countably infinite: this means their elements can be paired with natural-number positions, not that the list ends. An ellipsis requires a clear continuing rule.
The empty set ∅, also written {}, has no elements. The set {0} has one element, the number zero. The set {∅} also has one element, which happens to be the empty set itself. For positive integers x with x<1, the resulting set is empty. Zero is not a spare symbol to put inside an empty roster.
For A={p,q}, all subsets are ∅,{p},{q},{p,q}. Every set contains ∅ as a subset because there is no element of ∅ that could violate the subset condition. The opening source illustration lists only three nonempty subsets, but its later worked example includes ∅; the complete list needs it. A proper subset is a subset different from the whole set. A finite n-element set has 2ⁿ subsets and 2ⁿ−1 proper subsets, since each element is either included or not.
The universal set U contains all elements under discussion; every set being considered is a subset of U. It is a context, not a set containing absolutely everything. For A={2,4}, U={1,2,3,4} is possible, but a larger stated universe could also contain A. A rectangle represents U in our Venn diagrams. A suitable universe is not necessarily the only possible choice.
With U={1,2,3,4,5,6,7,8} and A={2,4,6,8}, Aᶜ=U∖A={1,3,5,7}. The superscript c, or a prime mark A′ in some books, means complement relative to the chosen U. Changing U may change Aᶜ. Inside U, (Aᶜ)ᶜ=A, Uᶜ=∅ and ∅ᶜ=U. The set A and its complement share no elements and together contain all of U.
For U={1,2,3,4,5,6} and A={2,4}, the elements of U not in A are {1,3,5,6}. They form Aᶜ. Enlarging U can enlarge this complement without changing A.
It lists members, rather than being a numerical data table. Equivalent element order is acceptable.
Do not treat a source example’s possible universe as the only set capable of containing its named subsets.
The braces and their contents have different jobs.
| Written set | Elements | Number of elements |
|---|---|---|
| ∅ | None | 0 |
| {0} | The number zero | 1 |
| {∅} | The empty set as an element | 1 |
| {1,2,2,3} | The distinct numbers 1,2,3 | 3 |
An empty set is not a container that must be filled with the number zero. Repetition does not change membership.
Choose whether to include each of p and q.
| Include p? | Include q? | Subset |
|---|---|---|
| No | No | ∅ |
| Yes | No | {p} |
| No | Yes | {q} |
| Yes | Yes | {p,q} |
All four are subsets; only the first three are proper subsets. The complete list includes the empty set.
Here U={1,2,3,4,5,6,7,8} and A={2,4,6,8}.
Aᶜ = U∖A = {1,3,5,7} A∩Aᶜ = ∅ A∪Aᶜ = U (Aᶜ)ᶜ = A
A larger universe could add members to the complement. Never complement against an unstated collection.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original membership model, not copied source artwork. Circle areas and distances do not encode member counts. Read the labels and element rosters.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original membership model, not copied source artwork. Circle areas and distances do not encode member counts. Read the labels and element rosters.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original membership model, not copied source artwork. Circle areas and distances do not encode member counts. Read the labels and element rosters.
Show your working on paper. Check your answer, then choose Next.
3
Repeating 4 does not create another element.
1
Twelve is one number, not the two separate elements 1 and 2.
5
They are 4,8,12,16,20.
5
The roster is {5,7,9,11,13}; both endpoints are excluded.
6
They are 1,2,3,6,9,18.
0
The empty set contains no element.
1
It contains the single element zero.
1
It contains one element: the empty set.
4
Each of two elements has an include/exclude choice: 2²=4.
8
2³=8 includes both ∅ and the whole set.
7
Exclude the whole set from its eight subsets.
0
Its only subset is itself; no proper subset remains.
4
The complement is {1,3,5,7}.
4
The complement in this specific U is {1,3,5,6}.
2
Double complementation returns A, which has two elements.
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The distinct elements in each are exactly 2,4,6. Order and repetition do not change membership.
Compare elements in both directions rather than comparing written lengths.
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The first says a number is an element. The second says every member of the singleton set {2} belongs to A. They are different relations with different objects on the left.
Use the symbols with their full meanings.
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{5,10,15,20,25}={x∈N:5 divides x and 1≤x≤25}, with N={1,2,3,...}. Other equivalent bounded rules are acceptable.
State the domain, positivity and upper bound.
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The pairs 1×18,2×9,3×6 give {1,2,3,6,9,18}. Each positive divisor appears once.
Keep positive divisors distinct from prime divisors.
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For example {x∈Z:1≤x≤4} gives {1,2,3,4}. Favourite numbers needs a specified person and a stated list or criterion before its membership can be decided.
Explain what additional information would make the unclear collection precise.
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Every natural number has a successor, so there is no last element. The list 1,2,3,... assigns a position to each number and is countably infinite; it does not finish after some large finite count.
Do not confuse an uncompleted count with proof that a finite set is infinite.
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∅ has no members. {0} contains one number, zero. {∅} contains one set, the empty set. Thus their counts are 0,1,1, and the three sets are distinct.
Identify the actual element when the count is one.
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The subsets are ∅,{p},{q},{p,q}. The proper subsets are the first three; the whole set is excluded from the proper list.
Include the empty subset, which the source opening illustration leaves out.
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For each of the n distinct elements choose in or out. Independent choices produce 2×2×...×2=2ⁿ selections, including all-out and all-in. For n=0 the empty choice gives the one subset ∅.
Explain the choices, not only the formula.
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They are ∅,{1},{2},{3},{1,2},{1,3},{2,3}. There are seven; the full three-element set is not proper.
Check for omissions, repeats and accidental inclusion of the whole set.
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For example U₁={1,2,3,4} and U₂={1,2,3,4,5,6}. Both contain A, but its complements are {1,3} and {1,3,5,6}.
State each universe explicitly and compute within it.
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Aᶜ={1,3,5,7}. Draw U as a rectangle, A inside it, and place these four odd elements outside A but inside U.
No complement element may be placed outside the universe.
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An element in A is outside Aᶜ and returns on the second complement. An element of U outside A enters Aᶜ and is removed by the second complement. Thus membership matches A exactly.
Use both membership cases and a fixed U.
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It admits zero and negative evens as well as evens not divisible by 4. Specify a positive natural number, divisibility by 4 and x≤20. The complete rule determines the intended list.
Name each missing restriction rather than silently imposing it.
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No. At a fixed time the school has a finite number of enrolled learners even if you do not know the count. Unknown information differs from an unending mathematical collection. The school-and-date context defines the universe.
Keep the mathematical meaning separate from what the observer happens to know.
Build a set-notation card collection and a complete subset tree for a three-element set.
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This original Kidemy practice follows the topic order of 2026 Class 8 Mathematics · English Version, chapter 7, printed pages 121–132. It does not replace the complete chapter or its practical teaching.
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