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Closed Sets

A set is considered closed for a particular operation (addition, subtraction, multiplication, or division) if doing that operation on any two numbers in the set produces another number that's already in the set.

That is certainly a mouthful. But, once we break it down a little bit, the idea of a closed set isn't too bad.

First, let's briefly review what a set is.

Set is a word that has basically the same meaning in math as it does in regular life. A set is a collection or a bunch - a list or grouping of distinct objects (what we usually call "elements" in math).

So, let's return now to the idea of a closed set.

Let's use the set below as an example:

Let's see if the set is closed under addition. In order to be closed, when we add together any two numbers in our set, we must only get values that are already in the set.

If adding two of our numbers together produces a value that falls outside our set, then our set is NOT closed.

This single example might make us think that our set is closed under addition. However, it's critical that you try more than one example.

You know, in court, you're innocent until proven guilty? Well, in our case, assume sets are open until proven closed.

Since we've found a value outside our original set, we can say that the set is NOT closed under addition.

Let's try a different example with a new set. This time, we'll see if our set (below, in yellow) is closed under multiplication.

Okay, so this might lead us to believe that our set is closed under multiplication. But, remember: open until proved closed.

We need to try using 2 different values from our set.

No matter what 2 values we multiply together in our set, we'll always get a number that is already in the set. Because of this, we can say that our set is closed under multiplication.

(Video) Closure Properties for Math

by HCCMathHelp

In this video, HCCMathHelp discusses closure properties. For example, is the set of natural numbers closed under addition? This means when you add natural numbers, is the result a natural number?

$\large{\text{Summary}}$

A closed set is made up of numbers that are the sums (or differences, or quotients, or products) of other numbers in the set. If any operation on any numbers in the set results in a number outside the set, then the set is no longer closed.

• Natural numbers are closed under addition.
• Natural numbers are NOT closed under subtraction.
• Natural numbers are closed under multiplication.
• Natural numbers are NOT closed under division.

A set is closed if and only if an operation on any two elements of the set produces another element of the same set. If the operation produces even one element outside of the set, the operation is not closed.

Example

The set of all real numbers $\mathbb{R}$ is closed under addition and multiplication. If you add or multiply any two real numbers you will get another real number. $$\begin{align} 3+4&=7\\ 6.7+8.1&=14.8\\ \sqrt{5}+3\sqrt{5}&=4\sqrt{5}\\ 2 \cdot 30 &= 60\\ 3.7 \cdot 4.3 &= 15.91\\ \sqrt{15} \cdot \sqrt{60} &= 30 \end{align}$$