Why the Empty Set Is a Subset of Every Set

Relations, functions, probability and calculus are all written in the language of sets. A set is simply a well-defined collection of objects, but the notation for describing, comparing and combining sets appears in almost every chapter that follows.

This lesson covers ways of writing sets and their types, subsets, intervals and power sets, operations on Venn diagrams, and the laws of complements.

A set is written in roster form by listing its elements inside braces, or in set-builder form by stating the property its elements share; it is empty if it has no elements, finite if its elements can be counted to an end, and infinite otherwise.

Two forms:

- Roster form — $A=\{1,2,3,4,6,12\}$, the divisors of 12; order does not matter and repeated elements are written once
- Set-builder form — $A=\{x:x\in \mathbb{N}\text{ and }x\text{ divides }12\}$

Types of sets:

- Empty set — $\varphi$, for example $\{x:x\in \mathbb{R},x^2=-4\}$
- Finite set — the letters of the word MATHEMATICS, $\{M,A,T,H,E,I,C,S\}$, with $n(A)=8$
- Infinite set — $\{x:x\in \mathbb{R},0<x<1\}$

Worked example. Write $B=\{x:x^2-5x+6=0\}$ in roster form. Factorising, $(x-2)(x-3)=0$, so $B=\{2,3\}$ and $n(B)=2$.

An everyday example. The names in a class attendance register form a finite set listed in roster form.

The substance. **$\{0\}$ is not the empty set** — it contains one element, the number zero.

**Subsets of the real numbers are written as intervals — open, closed or half-open depending on whether the end points are included — and the power set of A is the set of all subsets of A, which has $2^n$ elements when A has n elements.

Subsets.** $A\subseteq B$ when every element of A is in B. The empty set is a subset of every set, because it has no element that could fail to belong.

Intervals for real numbers $a<b$:

- Open $(a,b)=\{x:a<x<b\}$
- Closed $[a,b]=\{x:a\le x\le b\}$
- Half-open $[a,b)=\{x:a\le x<b\}$

Worked example. For $A=\{1,2,3\}$:

$$P(A)=\{\varphi ,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$$

so $n(P(A))=2^3=8$. A set with 5 elements has $2^5=32$ subsets, of which $31$ are proper.

An everyday example. A thali menu of dal, rice and sabzi can be ordered in 8 different selections, from none of the dishes to all three — the power set of the menu.

The substance. **Write $1\in A$ but $\{1\}\subseteq A$** — confusing 'is an element of' with 'is a subset of' is the commonest slip.

**In a Venn diagram the universal set U is a rectangle and sets are circles inside it; the union $A\cup B$ shades everything in either circle, the intersection $A\cap B$ only the overlap, the difference $A-B$ the part of A outside B, and the complement $A^{\prime }$ everything in U outside A.

Operations:**

- $A\cup B=\{x:x\in A\text{ or }x\in B\}$
- $A\cap B=\{x:x\in A\text{ and }x\in B\}$; if it is $\varphi$, A and B are disjoint
- $A-B=\{x:x\in A,x\notin B\}$
- $A^{\prime }=U-A$

Counting formulae:

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$

$$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)$$

Worked example. In a class of 60 students, 35 like cricket, 28 like football and 12 like both. Then

$$n(C\cup F)=35+28-12=51$$

so $60-51=9$ students like neither, and $35-12=23$ like only cricket.

An everyday example. A housing society survey of families reading Hindi and English newspapers is sorted exactly this way.

The substance. Adding n(A) and n(B) counts the overlap twice — that is why $n(A\cap B)$ is subtracted.

The complement of a set obeys the complement laws, De Morgan's laws and the double complement law, and these let a long set expression be replaced by a shorter equivalent one.

Properties of the complement:

- Complement laws — $A\cup A^{\prime }=U$ and $A\cap A^{\prime }=\varphi$
- De Morgan's laws — $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$ and $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$
- Double complementation — $(A^{\prime }{)}^{\prime }=A$
- Empty and universal sets — ${\varphi }^{\prime }=U$ and $U^{\prime }=\varphi$

Worked example. Let $U=\{1,2,3,4,5,6,7,8\}$, $A=\{2,4,6,8\}$ and $B=\{1,2,3,4\}$. Then $A\cup B=\{1,2,3,4,6,8\}$, so $(A\cup B{)}^{\prime }=\{5,7\}$. Separately, $A^{\prime }=\{1,3,5,7\}$ and $B^{\prime }=\{5,6,7,8\}$, giving $A^{\prime }\cap B^{\prime }=\{5,7\}$ — De Morgan's law checks out.

An everyday example. 'Not tea or coffee' at a canteen counter means 'no tea and no coffee' — De Morgan's law in plain words.

The substance. A complement always depends on the universal set — change U and $A^{\prime }$ changes, even if A stays the same.