If the universal set, 𝑈, is the set of factors of the number 30, what is the complement of the empty set?
Let’s use a diagram to help us sort this out. If inside this rectangle is the universal set, then inside this box, we need to put the factors of the number 30. The factors of the number 30 are numbers that multiply together to equal 30. One times 30 equals 30. That means both the number one and the number 30 are factors of 30. Two times 15 equals 30. Two and 15 are factors. Three times 10 equals 30. Three and 10 are factors. And finally, five times six equals 30. Five and six are factors.
So here, we have our universal set with all the factors of the number 30. Inside the universal set, there’s something else. It’s the empty set or the null set. There’s a space inside the universal set with nothing in it. But our question doesn’t want to know what’s in the empty set. It wants to know what the complement of the empty set is.
The complement of any set is the elements in the universal set, but not in the indicated set. The complement of the empty set is the elements in the universal set, but not in the empty set. We know that we’re dealing with a complement because of the little mark above the empty set.
Here’s what that looks like. Everything highlighted in blue is part of the universal set, but not part of the empty set. That means the complement of the empty set includes all factors of the number 30: one, two, three, five, six, 10, 15, 30.