A, B, and L are three sets. L is the set containing positive multiples of three less than 25. A is the set containing the numbers six, 12, 18, and 24. B is the set containing the numbers three, 12, 15, 18, and 21. Part a) Fill this information in on the Venn diagram.
The two circles on our Venn diagram must represent the two sets given. They are A and B. The set L has been shown to represent the enclosing rectangle. L is sometimes called the universal set. That represents all possible elements that could be in our Venn diagram. Here, that universal set is all positive multiples of three less than 25.
Notice how the question specifies that they are positive multiples of three less than 25. That means numbers greater than zero and less than 25. Let’s begin by listing those numbers out.
The multiples of three are the numbers in the three times table. The first eight are three, six, nine, 12, 15, 18, 21, and 24. Before we add these to our Venn diagram, let’s look at the numbers that sets A and B have in common. A and B have the numbers 12 and 18 in their lists. These therefore go in the overlap between the two circles. That tells us that they occur in both sets A and B. In fact, the mathematical word for that is the intersection of sets A and B.
The intersection can be represented with this notation. It’s A followed by a letter that looks a little bit like the letter N followed by B. That just means A intersection B.
We can also cross the numbers 12 and 18 off of the list for all numbers in the universal set. The remaining numbers in set A are six and 24. We can also cross six and 24 off of our list of numbers in the universal set. The remaining numbers for set B are three, 15, and 21. And of course, we then cross them off of our list for all the numbers in set L.
Now we can go back to our list of numbers in the universal set. It was positive multiples of three less than 25. We’ve crossed off all the positive multiples of three less than 25 that we’ve already used. The only one left over is the number nine. This goes outside of the circles. That tells us it’s still included in the set L, which was positive multiples of three less than 25, but it’s not in the set of numbers for A and B. Once we are sure that we’ve included all numbers in set L, our Venn diagram is complete.
A number is chosen randomly from the set L. Part b) What is the probability that the number is in the set A intersection B?
We already specified that this little letter, the one that looks a bit like the letter N, represents the intersection of the two sets. That’s the overlap. The probability of an event occurring is found by dividing the number of ways that event can occur by the total number of outcomes.
To find the probability that a randomly chosen number is in the set A intersection B then, we need to find the number of ways this can happen. There are two numbers in the overlap, in the intersection of A and B. So the number of ways this event can happen is two. We said that there were eight positive multiples of three less than 25, so the total number of outcomes here is eight. The probability of choosing a number that lies in the intersection of A and B then is two-eighths.
We should always try to simplify our fractions where possible. Here, both two and eight can be divided by two. Two divided by two is one, and eight divided by two is four. We have shown that the probability the randomly chosen number lies in the intersection of A and B is one-quarter.