Video: Identifying Nonempty Sets

Which of the following choices is NOT an empty set? [A] The set of odd numbers between (but not including) 5 and 6 [B] The set of whole numbers between (but not including) 5 and 6 [C] The set of odd numbers between (but not including) 5 and 27 [D] The set of even numbers between (but not including) 5 and 6

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Video Transcript

Which of the following choices is not an empty set? The set of odd numbers between, but not including, five and six. The set of whole numbers between, but not including, five and six. The set of odd numbers between, but not including, five and 27. Or, the set of even numbers between, but not including, five and six.

Another word for an empty set is a null set. As the name describes, this is a set that doesn’t have any elements in it. It’s empty. The question gives us four possible choices. Three of them are empty sets. We need to find the choice that is not an empty set. Which set contains numbers? Let’s go through our choices one by one.

Our first set is the set of odd numbers between, but not including, five and six. The phrase but not including is important here. We can’t include five or six in our set. So, how many odd numbers are there in between the numbers five and six? Well, there aren’t any numbers. Five and six are consecutive numbers. They’re next door to each other. There aren’t any odd numbers in between. And so, we know that this first set is going to be an empty set.

In our second choice, we’re given some numbers in between five and six again, the set of whole numbers between, but not including, five and six. So again, we can’t include the numbers five and six in our answer. How many whole numbers are there in between five and six? Well, for the same reasons as before, there aren’t any whole numbers. Five and six are numbers that are next to each other. The only numbers in between are decimals. So, the set of whole numbers between, but not including, five and six is another empty set.

For our third choice, we’re going to need to sketch a longer number line, the set of odd numbers between, but not including, five and 27. This number line shows values from five to 30. And we need to consider all the odd members between five and 27. So, let’s mark those two numbers on our number line. Here’s number five, and here’s where the number 27 is. Now as with all the other choices, we’re not allowed to include these numbers in our set.

But there are some other odd numbers that we can include, all the odd numbers that belong in between five and 27. What are they? They’re the numbers seven, nine, 11, 13, 15, 17, 19, 21, 23, and 25. This set contains 10 different numbers or 10 elements. The set of odd numbers between, but not including, five and 27 is a set that includes the elements seven, nine, 11, 13, 15, 17, 19, 21, 23, and 25. This choice is not an empty set.

So, although we think we’ve found the answer, let’s just check the final choice. And we’re back here to thinking of numbers in between five and six again, the set of even numbers between, but not including, five and six. And, of course, there are no even numbers in between five and six if we’re not allowed to include six. So, this set is an empty set. The set that is not an empty set is the one that contains 10 different elements. And this is the set of odd numbers between, but not including, five and 27.

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