The empty set, finite and infinite sets

How big is a set? Sometimes the answer is "zero". Sometimes it is a definite number. Sometimes the set goes on forever. This subtopic gives precise names to all three situations and shows why each one matters.

Definitions

A set is called the empty set (or null set or void set) if it has no elements. It is denoted by $\varnothing \quad \text{or}\quad \{\,\}.$ There is only one empty set, even though it can be described in many ways. For instance $\{x \in \mathbb{R} : x^2 + 1 = 0\} = \{x \in \mathbb{N} : x < 0\} = \{x : x \text{ is a girl studying in a boys' school}\} = \varnothing.$

A set $A$ is finite if its elements can be counted out , formally, if there is a natural number $n$ such that the elements of $A$ are exactly $a_1, a_2, \dots, a_n$. The number $n$ is called the cardinality of $A$, written $|A|$ or $n(A)$. By convention $|\varnothing| = 0$.

A set is infinite if it is not finite. The set $\mathbb{N}$ of natural numbers is infinite; so are $\mathbb{Z}, \mathbb{Q}, \mathbb{R}$.

Why "empty set" matters

The empty set is the additive identity of set algebra: $A \cup \varnothing = A$, $A \cap \varnothing = \varnothing$. It is also the set of solutions whenever an equation or inequality has no solutions , and you need a notation for that. Writing $\{x \in \mathbb{R} : x^2 < 0\} = \varnothing$ is much sharper than "no real numbers satisfy this".

A subtle point: $\varnothing \ne \{\varnothing\}$. The first has zero elements; the second has one element (and that element is itself the empty set). Building sets out of sets is how higher mathematics is constructed; we will see it again with the power set in a later subtopic.

Finite vs infinite

A finite set has a definite size: $\{1,2,3\}$ has $3$ elements. We can list its elements completely.

An infinite set cannot be listed completely. Examples:

• $\mathbb{N} = \{1, 2, 3, \dots\}$
• The set of all even integers
• The interval $(0, 1) = \{x \in \mathbb{R} : 0 < x < 1\}$
• The set of all points on a line
• The set of all primes (Euclid proved there are infinitely many)

Some finite sets look large but have a precise size. The set of seven-digit positive integers has $9 \cdot 10^6 = 9{,}000{,}000$ elements , still finite. By contrast, $\{x \in \mathbb{R} : 1 \le x \le 1.0001\}$ is infinite because between any two real numbers lie infinitely many others.

Equivalent vs equal sets

Two finite sets $A$ and $B$ are equivalent if $|A| = |B|$ , same size, possibly different elements. They are equal if they have exactly the same elements. Equal sets are always equivalent; the converse is false: $A = \{1,2,3\},\quad B = \{a, b, c\}: \quad |A| = |B| = 3,\quad \text{yet } A \ne B.$

(For infinite sets, the notion of "size" needs more care; you will meet it in higher classes. For this chapter we stay with finite cardinalities.)

Worked examples

Example 1. Find the cardinality of $A = \{x \in \mathbb{N} : x \le 10,\ x \text{ is prime}\}$.

Primes up to $10$ are $2, 3, 5, 7$, so $A = \{2,3,5,7\}$ and $|A| = 4$.

Example 2. Is $\{x \in \mathbb{R} : x^2 - 4x + 5 = 0\}$ empty?

Discriminant $= 16 - 20 = -4 < 0$, so the quadratic has no real roots. The set is $\varnothing$.

Example 3. Decide finite or infinite: (i) set of all months of a year; (ii) set of all positive integers less than $100$; (iii) set of all multiples of $5$; (iv) set of all rational numbers between $0$ and $1$.

(i) Finite, $12$ elements. (ii) Finite, $99$ elements. (iii) Infinite. (iv) Infinite (between any two rationals lies another).

Example 4. How many elements are in $\{x \in \mathbb{Z} : -7 < x < 4\}$?

The integers strictly between $-7$ and $4$ are $-6, -5, \dots, 3$, which is $3 - (-6) + 1 = 10$ elements.

Example 5 (harder). Let $A = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 = 25\}$. Find $|A|$.

We need integer pairs with $x^2 + y^2 = 25$. Possible $(|x|, |y|)$: $(0, 5), (3, 4), (4, 3), (5, 0)$. Each non-zero coordinate gives two signs:

• $(0, \pm 5)$: $2$ pairs.
• $(\pm 3, \pm 4)$: $4$ pairs.
• $(\pm 4, \pm 3)$: $4$ pairs.
• $(\pm 5, 0)$: $2$ pairs.

Total: $|A| = 12$.

Try it yourself

1. Which of these are empty? (a) $\{x \in \mathbb{R} : x^2 = -9\}$; (b) $\{x \in \mathbb{R} : x = x + 1\}$; (c) $\{x : x \text{ is a prime less than } 1\}$; (d) $\{x \in \mathbb{Z} : -2 < x < -1\}$.
2. Find the cardinality of $\{x \in \mathbb{N} : x \le 20,\ x \text{ is divisible by } 3\}$.
3. Finite or infinite? (a) $\mathbb{Q}$; (b) lines through the origin in the plane; (c) digits in the decimal expansion of $\pi$; (d) points on the segment from $(0,0)$ to $(1,1)$.
4. Find $|A|$ where $A = \{x \in \mathbb{Z} : x^2 < 36\}$.
5. Show that the set of all factors of $36$ has size $9$.
6. How many three-digit positive integers are there?
7. Find $|\{(x, y) \in \mathbb{Z}^2 : |x| + |y| = 2\}|$.
8. Why is $\{x \in \mathbb{R} : \sin x = 2\}$ empty?
9. Distinguish $\varnothing$, $\{0\}$, $\{\varnothing\}$. How many elements does each have?
10. Find the cardinality of $\{x \in \mathbb{N} : x \text{ divides } 60\}$.
11. Are $\{1,2,3,4\}$ and $\{a,b,c,d\}$ equal? Equivalent?
12. List two examples each of: empty sets, finite sets of size $7$, infinite sets.

Pitfalls / Tricks

• $\{0\}$ is not the empty set , it contains one element, namely $0$.
• $\varnothing \subseteq A$ for every set $A$ (vacuously). We will use this constantly.
• Never write $\{\varnothing\}$ when you mean $\varnothing$. The braces matter.
• Insight. When asked "is this set finite?", first ask "can I in principle list every element?". If between any two elements lies another, the set is infinite.