How to Define a Set

We know, from the previous article, what is a set. In this article we’re going to learn how to define one. What’s wrong with the simple way of defining a set by enumerating all its elements? Like this one: A = {a, b, c}. Because it’s hard to define more complex sets this way.

Instead of enumerating we can use another approach. We can define a set by defining a characteristic property of elements of a set. Take a look at the example:

Let the set S be equal to all n2 for n ∈ ℕ.

In other words, the set consists of all squares of all natural numbers. We can enumerate it this way: S = {1, 4, 9, 16, 25, …}. This definition is quite ok but we can go deeper and we can cut the words and we can replace them by some math symbols:

S = {n2|n∈ℕ}.

This definition is exactly the same. Inside of curly brackets we have two expressions separated by the “|” symbol. The right one tells us which n are we enumerating and the left one tells us what we are doing with those numbers. In this case we enumerate natural numbers and we take square of each number. Another example:

Let the set S be equal to all x ∈ ℝ that are greater than zero.

The definition is clear. How can we rewrite it in term of mathematical symbols? Simply:

S = {x|x∈ℝ and x > 0}

The definition says we take all real numbers that are greater than zero and we do nothing with them. Let’s take another approach. Let’s take this set:

S = {x|x · x = x for x∈ℝ}.

We are looking for all real numbers that we can multiply by themself and result is this number again. There are only two numbers that satisfy the condition and it’s S = {0, 1}. Because 0 · 0 = 0 and 1 · 1 = 1.