Thanks to Kellie D for this question: “How do I know if a set is closed under the rule of addition, same question for multiplication?”
Closure under addition or multiplication is a concept from Abstract Algebra that’s taught in not-so-abstract Algebra classes. It can be intuitive, if taught properly.
You first have to start with two things: a set and an operation. A set is a bunch of things. Okay, the definition is fuller than that, but for general purposes, it’s a basket full of goodies.
An operation is a way to “combine” any two things in that basket.
You can’t speak of closure without having both of these together. In Abstract Algebra, this pair of things is called an Algebraic System. That’s a fancy math term that isn’t necessary to teach, but might be helpful if you need to look things up.
We say a set is closed under an operation if you can combine any two things and the result is back in the set. Notice that the definition has BOTH the words set and operation. You can’t just be closed under an operation – you gotta have the set to put it in context.
Example 1 – Kitchen Goodies
Suppose your basket of goodies is this: {flour, sugar, chocolate chips, baking powder, egg, butter}.
And your operation is this: “mix two things together in a bowl with a spoon.”
It isn’t mathematical, per se, but it’s a set and an operation.
So if you combine two things from the basket by mixing them together, what do you get? Yummy goodness for sure, right? But the result of the blend is not found in the basket as one of the things!
Mix sugar with butter and then look in the basket. Is it there as a single blob of buttery sweetness? Nope. So the set of goodies is not closed under the operation of mixing.
Note: the opposite of closed is “not closed.” The word “open” isn’t used here.
Example 2 – Math Stuff
Now you have another basket of goodies: all the even numbers (including positives and negatives and zero). We can write this in the fancy math way like this: { x | x is an even integer }
The operation is addition.
If you combine (add) two things from the basket (any two even numbers), what do you get?
8 + 24 = 32
Is it back in the basket, i.e. is the answer an even number? YES!
Try some more to make sure. Technically you should prove it, but usually your intuition is good enough – especially in a high school or undergraduate class.
So we write “the set of even numbers is closed under the operation of addition.” Again, notice that we have BOTH set and operation involved.
Example 3 – More Math Stuff
Here’s your set: all the integers (positive, negative and zero whole numbers).
And the operation: Division.
If you combine (divide) two things from the basket (integers), what do you get?
27 ÷ 3 = 9
Looks good, right? 9 is an integer. But what about 27 ÷ 2 = 13.5?
13.5 isn’t an integer because it has a decimal piece. Some bits (integers) when combined (divided) turn out to be in the basket (another integer). But not all do. And they ALL have to work in order for it to be called closed.
So this is NOT a closed system. Or we can write, “the set of integers is not closed under the operation of division.”
Example 4 – Even More Math Stuff
Set: {1, 5, 6, 7, 14}
Operation: addition.
Is it closed?
1 + 5 = 6 check!
5 + 6 = 11 oops! 11 isn’t in that set.
This set is not closed under the operation of addition. You might notice that a finite set (a set with only so much stuff) of regular numbers won’t ever be closed under addition – just add the two biggest numbers and you “bust” out of the set.
Going Beyond…
Your sets can be just about anything: real numbers, rationals (fraction-like numbers), whole numbers. They can even be bunches of matrices, polynomials, imaginary numbers, complex numbers or functions! The operations would be one of the normal operations that you would use on things in that set anyway.
Questions about how to teach this? Questions on how to learn it? Leave them in the comments.
