A Set of Tall Students Is Not a Set at All

Because "tall" means different things to different people, so nobody can say for certain whether a given student belongs. A set must be well-defined — for every object, the answer to "is it in?" has to be a definite yes or no.

That one requirement is what makes sets usable in mathematics. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: well-defined sets and membership, roster and set-builder form, cardinal numbers and types of set, and equal versus equivalent sets.

A set is a well-defined collection of distinct objects, called its elements or members. Well-defined means membership is never a matter of opinion.

Well-defined, so these are sets:

- the vowels of the English alphabet
- the natural numbers less than 10
- the months of the year with 30 days

Not well-defined, so these are not sets:

- the tall students of a class
- the good cricket players in India
- the interesting chapters of this book

Each one depends on personal judgement, and good or interesting has no fixed test.

Sets are named with capital letters and written inside curly brackets. Membership uses two symbols:

- $\in$ means is an element of
- $\notin$ means is not an element of

So if $A=\{2,4,6,8\}$, then $4\in A$ and $5\notin A$.

Two rules about listing matter. Elements are never repeated — the set of letters of the word book is $\{b,o,k\}$, with three elements, not four. And the order does not matter, so $\{2,4,6\}$ and $\{6,2,4\}$ are the same set.

Roster (tabular) form lists every element inside curly brackets, separated by commas. Set-builder (rule) form states the property the elements share.

The same set, both ways:

$$A=\{2,4,6,8,10\}$$

$$A=\{x:x\text{ is an even natural number, }x\le 10\}$$

The colon is read as such that.

More conversions:

- $B=\{a,e,i,o,u\}$ becomes $\{x:x\text{ is a vowel of the English alphabet}\}$
- $C=\{x:x\text{ is a natural number, }3<x<8\}$ becomes $\{4,5,6,7\}$
- $D=\{1,4,9,16,25\}$ becomes $\{x:x=n^2,n\text{ is a natural number, }n\le 5\}$

A shop's price list is roster form; "everything under ₹100" is set-builder form describing the same shelf.

Set-builder form is the only practical choice when a set is infinite or very large. The natural numbers cannot be listed in roster form at all, so they are written $\{x:x\text{ is a natural number}\}$ — and writing $\{1,2,3,\dots \}$ with dots is a shorthand for exactly that rule.

The cardinal number $n(A)$ is the number of distinct elements in set $A$.

If $A=\{1,2,3,4,5\}$ then $n(A)=5$. If $B$ is the set of letters of the word mathematics, the distinct letters are $\{m,a,t,h,e,i,c,s\}$, so $n(B)=8$ — repeated letters are counted once.

Sets are then classified by that count:

- Finite set — the counting ends. $\{2,4,6\}$, with $n=3$.
- Infinite set — the counting never ends, such as the set of natural numbers.
- Empty (null) set — no elements at all, written $\{\}$ or $\varnothing$, with $n=0$. An example is $\{x:x\text{ is a natural number, }x<1\}$.
- Singleton set — exactly one element, with $n=1$. An example is $\{x:x\text{ is an even prime number}\}$, which is $\{2\}$.

The distinction that catches everyone is between $\varnothing$ and $\{0\}$. The empty set contains nothing, so $n(\varnothing )=0$. But $\{0\}$ contains the number zero — one element — so it is a singleton with $n=1$. Likewise $\{\varnothing \}$ is a singleton, not an empty set, because it has one thing inside it.

Equal sets have exactly the same elements. Equivalent sets have the same number of elements, whatever those elements are.

So with $A=\{1,2,3\}$, $B=\{3,2,1\}$ and $C=\{a,b,c\}$:

- $A$ and $B$ are equal, written $A=B$, since order does not matter.
- $A$ and $C$ are equivalent, since $n(A)=n(C)=3$, but they are not equal — no element is shared.

Every pair of equal sets is also equivalent, because identical elements means an identical count. The reverse is not true, which is the whole point of having two words.

Two further relationships are named by what the sets share:

- Disjoint sets have no common element. $\{1,2\}$ and $\{3,4\}$ are disjoint.
- Overlapping sets have at least one common element but are not equal. $\{1,2,3\}$ and $\{3,4,5\}$ overlap, sharing 3.

The pupils who play only cricket and those who play only football form disjoint sets; the pupils who play cricket and those who play football overlap, because some play both.

The test to apply is simply which question is being asked. Same elements? decides equal. Same count? decides equivalent. And a pair of equal sets can never be disjoint unless both are empty.