When Is a Collection a Set, and When Is It Just a Group?

A set is a well-defined collection of objects, which means there is a clear rule that decides whether any given object belongs to it or not. If two sensible people could disagree about whether something belongs, the collection is not a set.

"The vowels of the English alphabet" is a set — everyone agrees the answer is a, e, i, o, u. "The tall boys in my class" is not a set, because tall has no fixed boundary.

This page covers everything in the ICSE Class 6 Mathematics chapter on sets: deciding whether a collection qualifies, the three ways of writing a set down, the types a set can be, and the difference between equal and equivalent sets.

Apply one test: is there a rule that answers yes or no for every object, with no room for opinion?

These are sets: the days of the week, the even numbers between 1 and 20, the rivers of India, the letters of the word MATHS.

These are not sets: the good students of a school, the interesting books in a library, the difficult chapters in this subject. Each depends on someone's judgement, so membership shifts from person to person.

Objects in a set are called its elements or members, and each element is written only once. The collection of letters in the word BOOK has just three elements — B, O and K — because repeating O adds nothing.

For example, "the students in your class whose roll number is even" is a set, since the register settles every case. "The students who are good at cricket" is not.

The same set can be written three ways, and questions often ask you to convert between them.

Description (statement) form states the rule in words: A is the set of odd numbers less than 10.

Roster (tabular) form lists the elements inside curly brackets, separated by commas: $A=\{1,3,5,7,9\}$.

Set-builder form states the rule using a variable: $A=\{x:x\text{ is an odd number},x<10\}$, read as "the set of all x such that x is an odd number less than 10".

Two symbols describe membership. Use $\in$ for is an element of and $\notin$ for is not an element of. For the set above, $3\in A$ but $4\notin A$.

In roster form the order does not matter and repetition is not allowed, so $\{1,3,5\}$ and $\{5,1,3\}$ are the same set. That surprises students who expect a list to be ordered.

Sets are classified by how many elements they contain.

An empty (or null) set has no elements at all, written $\{\}$ or $\varphi$. The set of months with 32 days is empty.

A singleton set has exactly one element, such as the set of even prime numbers, which is $\{2\}$.

A finite set has a countable number of elements that comes to an end, such as the letters of the alphabet. An infinite set never ends, such as the set of all natural numbers.

The cardinal number $n(A)$ is simply how many elements set A has. If $A=\{2,4,6,8\}$ then $n(A)=4$. For the empty set, $n(\varphi )=0$.

A point worth pinning down: $\{0\}$ is not an empty set. It is a singleton containing the element 0, so its cardinal number is 1, not 0.

Equal sets contain exactly the same elements. Equivalent sets contain the same number of elements, but not necessarily the same ones.

Let $A=\{1,2,3\}$, $B=\{3,2,1\}$ and $C=\{5,6,7\}$.

A and B are equal, written $A=B$, because every element of A is in B and every element of B is in A — the different order changes nothing.

A and C are equivalent but not equal, since $n(A)=n(C)=3$ while their elements are entirely different.

For example, the set of your four classroom walls and the set of the four directions are equivalent — both have four elements — yet clearly not equal.

The relationship runs one way only: all equal sets are equivalent, because identical elements guarantee an identical count. But equivalent sets need not be equal.