A Set of Just 10 Elements Has More Than a Thousand Subsets

A set is a well-defined collection of distinct objects, called its elements. Well-defined means that for any object, you can say definitely whether it belongs.

Sets give mathematics a precise language for groups of numbers, points or outcomes. Every later topic — functions, probability, domains — is written in this language.

This part covers ways of writing sets, types of sets, subsets and power sets, and intervals of real numbers.

Roster form lists the elements inside braces, set-builder form describes the property shared by all elements, and a collection is a set only if membership can be decided without opinion.

Roster form: vowels of English $=\{a,e,i,o,u\}$.

Set-builder form: $\{x:x\text{ is a vowel of the English alphabet}\}$.

Worked example 1. Write $\{x:x\in \mathbb{N},x^2<30\}$ in roster form.

$$1^2,2^2,3^2,4^2,5^2=1,4,9,16,25<30\text{but}{6}^2=36\text{so }\{1,2,3,4,5\}$$

Worked example 2. $\{x:x=2n-1,n\in \mathbb{N},n\le 5\}=\{1,3,5,7,9\}$.

Worked example 3. The letters of SCHOOL form $\{S,C,H,O,L\}$ — repeated letters are written once.

Well defined or not?

- The tall students in a class — not a set, since tall is a matter of opinion
- **Students taller than $160$ cm — a set

An everyday example. The states of India that have a coastline form a well-defined set, because each state either touches the sea or does not.

The substance. Order and repetition do not matter**: $\{1,2,3\}$ and $\{3,1,2,2\}$ are the same set.

A set is empty if it has no elements, finite if its elements can be counted to an end, and infinite otherwise; two sets are equal exactly when they have the same elements.

Empty sets:

$$\{x:x\in \mathbb{N},1<x<2\}=\varnothing \{x:x\in \mathbb{R},x^2=-1\}=\varnothing$$

Finite and infinite:

- The days of the week — finite, with $7$ elements
- The natural numbers — infinite
- Points on a line segment — infinite

Equal sets. The letters of LOYAL and of ALLOY both give $\{L,O,Y,A\}$, so the two sets are equal.

Equal versus equivalent. $\{1,2,3\}$ and $\{a,b,c\}$ both have $3$ elements, so they are equivalent, but they are not equal, because their elements differ.

An everyday example. Two family chat groups with exactly the same members are equal sets, even if members joined in different orders.

The boundary case. **$\{0\}$ is not empty** — it has one element, the number $0$ — and $\{\varnothing \}$ is not empty either.

**$A$ is a subset of $B$ if every element of $A$ is in $B$; a set with $n$ elements has $2^n$ subsets, $2^n-1$ of them proper, and the set of all its subsets is its power set.

Worked example.** All subsets of $A=\{a,b,c\}$:

$$\varnothing ,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}$$

That is $2^3=8$ subsets, of which $2^3-1=7$ are proper, since $A$ itself is not a proper subset.

**Why $2^n$. Each element is either in or out** of a subset — two choices for each of the $n$ elements, so $2\times 2\times \cdots \times 2=2^n$.

Power set. $P(A)$ is the set whose elements are these $8$ subsets, so $n(P(A))=8$. For $n(A)=10$, $n(P(A))=2^{10}=1024$.

Chain of number sets: $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$.

An everyday example. **A thali with $5$ dishes** can be served as $2^5=32$ different selections, counting an empty plate and the full thali.

The trap. **$a\in A$ but $\{a\}\subseteq A$** — an element and a one-element subset are different things.

An interval is the set of all real numbers between two end points; round brackets exclude an end point, square brackets include it, and infinity always takes a round bracket.

- Open: $(a,b)=\{x:a<x<b\}$
- Closed: $[a,b]=\{x:a\le x\le b\}$
- Half-open: $[a,b)=\{x:a\le x<b\}$ and $(a,b]=\{x:a<x\le b\}$
- Unbounded: $(-\infty ,3]=\{x:x\le 3\}$ and $(2,\infty )=\{x:x>2\}$

Worked example 1. $\{x:-2<x\le 5\}$ is $(-2,5]$. On a number line: a hollow circle at $-2$, a filled circle at $5$, shaded between.

Worked example 2. $[3,7)$ means $3\le x<7$; its length is $7-3=4$.

Worked example 3. Solving $2x-1\ge 5$ gives $x\ge 3$, which is $[3,\infty )$.

An everyday example. **A luggage rule allowing bags up to and including $15$ kg** describes weights in the interval $(0,15]$.

The substance. **$(a,b)$ as an interval is not the ordered pair $(a,b)$** — context tells you which is meant.