Oct 15, 2026 · 7 min · 5 segments
We've already learned a little bit about set theory when we first started using interval notation. Let's expand on this a little bit to include sets that merely list some elements, so that we can…
As we have been learning algebra, we’ve been learning about different kinds of sets.
If any number can be plugged into a function, then the domain of that function is the set of all real numbers.
This can be represented by this parenthetical term of negative infinity comma positive infinity.
We use parentheses because they imply non-inclusivity, and we can never actually get to infinity, so it’s not included in the set.
If X is greater than three, the solution is the set of all real numbers greater than three, which would be three comma infinity in parentheses.
Again, since three is not included in the solution set, we use an open parenthesis to start.
If it was X is greater than or equal to three, then three would be included in the solution set, and this would instead be an open bracket symbol, which implies inclusivity.
We could have an open interval, which would be the set of all numbers in between A and B.
We could have a closed interval, which would be the set of all numbers between A and B, and including A and B themselves.
Intervals that include positive or negative infinity are called infinite intervals.
We can also have sets that are not continuous intervals, but rather just collections of elements.
These utilize this other type of bracket, and they just list all of the elements that are included in the set, separated by commas.
We could have a set like this, which includes the numbers one, four, five, nine, and thirteen.
Then let’s say we have another set like this, which includes two, five, six, eight, and nine.
One thing we could do would be to describe the intersection and the union between these two sets.
The intersection of two sets includes all the items that are common to both sets.
We can think of this as the part of a Venn diagram where the two objects overlap.
In this case, the items that are present in both sets are five and nine, so the intersection of these two sets, which we represent with this symbol that looks like an upside down capital U, will be the set containing five and nine.
Every item that shows up in either set will be in the union of the sets, so with the Venn diagram, the union is just the whole thing.
In this case, the union of these two sets, which is represented by this symbol that looks like a right side up capital U, will be a larger set containing all of these numbers.
For example, let’s find the intersection between the set containing all the odd natural numbers and the set containing all the even natural numbers.
As we have been learning algebra, we’ve been learning about different kinds of sets.
If any number can be plugged into a function, then the domain of that function is the set of all real numbers.
This can be represented by this parenthetical term of negative infinity comma positive infinity.
We use parentheses because they imply non-inclusivity, and we can never actually get to infinity, so it’s not included in the set.
If X is greater than three, the solution is the set of all real numbers greater than three, which would be three comma infinity in parentheses.
Again, since three is not included in the solution set, we use an open parenthesis to start.
If it was X is greater than or equal to three, then three would be included in the solution set, and this would instead be an open bracket symbol, which implies inclusivity.
We could have an open interval, which would be the set of all numbers in between A and B.
We could have a closed interval, which would be the set of all numbers between A and B, and including A and B themselves.
Intervals that include positive or negative infinity are called infinite intervals.
We can also have sets that are not continuous intervals, but rather just collections of elements.
These utilize this other type of bracket, and they just list all of the elements that are included in the set, separated by commas.
We could have a set like this, which includes the numbers one, four, five, nine, and thirteen.
Then let’s say we have another set like this, which includes two, five, six, eight, and nine.
One thing we could do would be to describe the intersection and the union between these two sets.
The intersection of two sets includes all the items that are common to both sets.
We can think of this as the part of a Venn diagram where the two objects overlap.
In this case, the items that are present in both sets are five and nine, so the intersection of these two sets, which we represent with this symbol that looks like an upside down capital U, will be the set containing five and nine.
Every item that shows up in either set will be in the union of the sets, so with the Venn diagram, the union is just the whole thing.
In this case, the union of these two sets, which is represented by this symbol that looks like a right side up capital U, will be a larger set containing all of these numbers.
For example, let’s find the intersection between the set containing all the odd natural numbers and the set containing all the even natural numbers.
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