Given that 𝑥 is the closed interval from negative six to three and 𝑦 is the set containing the elements negative four and three, find 𝑥 minus 𝑦.
In order to answer this question, we’re going to begin by considering the representation of these sets on a number line. 𝑥 is an interval from negative six to three, and the square brackets mean we include these values. On a number line, that looks a little something like this. These solid or filled-in circles at the end of our lines mean we include the numbers negative six and three as well as all of values between those.
These squiggly brackets for 𝑦 means something different, though. This means the set containing simply the elements negative four and three. We might choose to represent this on the number line as simply two circles. And the question wants us to find 𝑥 minus 𝑦, in other words, the values in 𝑥 minus the values in 𝑦. Well, if we think about what this might do to our number line, this is quite unconventional, but it looks like we have a break in our line.
We don’t want to include the number negative four or the number three since we’re taking away the values of 𝑦 from the interval 𝑥. And so, we’ll begin by considering this as two separate intervals. First, we have the set of values from and including negative six up to but not including negative four. We can put a square bracket on the left and a round bracket on the right to represent this.
Then, the next set of values is all values from negative four to three but not including negative four and three. And so, we use purely round brackets here. This is called the open interval from negative four to three. We want all of these values. So, we use the symbol that looks a little bit like a letter U. This is the union. And it tells us that we’re interested in all the values in both intervals plus any values in an overlap.
In interval notation, we can say that 𝑥 minus 𝑦 is the left-closed, right-open interval from negative six to negative four union with the open interval negative four to three.