Orders of Sets
Yup. That’s it. If given a set and asked for its order, just count the number of elements in the set and write that down!
But what if you’re dealing with something like the set of all integers \(\mathbb{Z}\)? What is \(|\mathbb{Z}|\)?
Try computing orders of the sets below in the examples!
Try the following problems on computing order!
\(|S|=4\)
Literally count the number of elements in the set and write that number
\(|A|=4\)
Be careful about what the elements of the set \(A\) actually are. Here, we are looking at the elements/objects in the set: 1,2, the set \(\{3,4,5\}\), and the set \(\{1,2\}\); i.e. these smaller sets are elements in the bigger one because they are single objects themselves; doesn’t matter what’s inside each of them because we are looking at these sets as a whole.
Think of having a box that contains three hats and two smaller boxes that are taped shut; one containing shirts, the other containing pants. How many objects are in the bigger box? In this case, 5; three hats and two boxes.
\(|B|=11\)
\(B\) is the set of all natural numbers that are strictly less than 12. How many numbers are between 1 and 11, including 1 and 11?
\(|C|=15\)
Here we are looking at the set of all natural numbers between 10 and 24, including 10 and 24. How many natural numbers have this property?
\(|2\mathbb{Z}|=\infty\).
Here we are looking at the set of all even integers. How many of those are there? If you at first thought there were finitely many evens, you’d find that if you try to list them all, you’d be going forever. Don’t let the weird combination of symbols \(2\mathbb{Z}\) scare you. Its just the name I gave the set. I could have called it “smiley face.”
\(|S|=\infty\)
\(S\) is the set of all INTEGERS less than 25, meaning we’re playing with negative stuff too! How many negative numbers are less than 25? Make sure you get ’em all if you try to list them out ;).
Infinite
Write out decimal numbers of the form 0.1, 0.11, 0.111, 0.1111, 0.11111, 0.111111,…. all of these numbers are indeed decimals between 0 and 1. I could forever add more numbers to my list just by tacking on more 1’s!
Infinite
0.1, 0.11, 0.111, 0.1111, 0.11111, 0.111111, 0.1111111, …. are all numbers inside this interval. This list of numbers is infinite because I can always add more 1’s to continue the pattern.
\(|A|=12\)
This is a game of carefully considering what elements live in \(A\), and it helps to write out as many, if not all, elements that meet the properties described in the set.
Here, we are looking at the set of all even natural numbers no greater than 24. How many of these sorts of elements are there?
Finite!
There is nothing in the empty set, so you cannot have an infinite number of elements in it.
Put another way, if you have no money – like zero dollars – would you say you have infinite cash? 😉