It’s Professor Dave, let’s learn about sets.
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.
We can use sets to express the solutions to inequalities.
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.
These solutions use interval notation.
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.
By contrast, the union of two sets is the merging of the sets.
Every item that shows up in either set will be in the union of the sets, so with this
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.
Sometimes we might have to deal with a set that has no elements at all.
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.
Because these sets have no elements in common, their intersection has nothing in it.
This will be the empty set, otherwise known as the null set.
This is represented by the Greek letter phi, which looks kind of like a zero with a line
through it.
Remember, this isn’t zero, because zero is an element that could be included in a set.
The null set means there is nothing in it, not even zero.
As we said, we can use interval notation to represent the solutions of inequalities, and
sometimes we will need unions and intersections to do this.
Say we are solving absolute value of X is greater than two.
Looking at this number line, any number greater than two or less than negative two will be
a solution, so the solution of the inequality will be the union of the real numbers from
negative infinity to negative two and from two to positive infinity.
This is the best way to depict solutions of this nature.
We could represent solutions to systems of inequalities this way as well.
Say we have X is greater than negative three and X is less than or equal to four.
Again, looking at the number line, anything greater than negative three works for the
first one, and anything less than or equal to four works for the other one, but the only
values that will be solutions to the system of inequalities will be values that are solutions
for both inequalities.
That will be this region of overlap here, so the solution to the system will be the
interval from negative three exclusive to four inclusive.
It is important that we open with a parenthesis and close with a bracket, as that will accurately
reflect the boundaries of the solution set.
Set theory gets more in depth than this, but that’s all we really need to know for algebra,
so let’s check comprehension.