Finite intervals in greater detail
Finite intervals are further divided according to where they are closed.
Closed interval
A closed interval is a kind of interval, which is closed on both sides with given values belonging to the interval. It means that there are square brackets on both sides of the interval, e.g. [1,5].
Note down the interval [1,5] using set-builder notation as 1 \leq x \leq 5 (read: "number x is greater than or equal to one, and, at the same time, less than or equal to five"). In other types of intervals you should focus on the changing sign of inequality (i.e. "<" and "\leq") while using the set-builder notation.
You can graph this type of interval on a real number line. Graph both endpoints on the line with filled-in circles, because they belong to interval, indicated by square brackets \rightarrow[1,5]. If the points were not to be included, which means there would be round brackets at the endpoints, then they would be graphed by open circles on the real number line, but we'll get to that later.
b) Right-closed interval
The right-closed interval (and left-open interval, depends on how you look at it) is the kind of interval, which is closed on the right side (the square bracket indicates that the endpoint belongs to the interval) and open on the left side (the round bracket indicates that the endpoint doesn't belong to the interval), e.g. interval (-1,2].
Note down the interval (-1,2] using set-builder notation as -1<x\leq2 (read: "number xis greater than negative one, and, at the same time, less than or equal to two"). As you can see, if the bracket at the endpoint is round, you will use the "<" sign. If there is square bracket in the assignment, you will use the \leq sign.
This interval is noted by drawing a filled-in circle on the real number line over the value of the endpoint with the square bracket. An open circle will be over the endpoint with the round bracket. The real number line below will tell you more.
A square bracket in assignment of the interval and a filled-in circle on the real number line indicate, that this point belongs to the interval. Whereas round bracket in the assignment of the interval and open circle on the real num-ber line says, that this point doesn't belong to the interval, the real number line above will tell you more.
c) Left-closed interval
A left-closed interval (also right-open interval) is a kind of interval, which is closed on the left side (square bracket) and open on the right side (round bracket), e.g. interval [0,3).
Note down the interval [0,3) using set-builder notation as 0 \leq x<3 (read: "number x is greater than or equal to zero, and, at the same time, less than three"). Where there is a round bracket, put "<" and where there is a square bracket, write "\leq".
You can of course graph this type of interval on the real number line. The endpoint limiting the interval from the left belongs to the interval (therefore, it's marked on the line with a filled-in circle), while the endpoint, which limits the interval from the right, doesn't belong to the interval (therefore, it's marked on the line with an open circle).
d) Open interval
An open interval is a kind of interval, which is opened on both sides (only round brackets), e.g. (-2,2). Note down the interval (-2,2) using set-builder notation as -2<x<2 (read: "number x is greater than negative two, and, at the same time, less than two").
You can graph this type of interval on the real number line by marking both endpoints as open circles because they don't belong to the interval. The interval is denoted with round brackets.
