Union and Intersection Symbols with Multiple Sets

means the number of elements in the intersection of set ‘S’ and set ‘R’.  In other words, the number of elements which are part of both sets ‘S’ and ‘R’.  But we can also have expressions which have more than just two sets.  For instance, take a look at this one:

In the last question, ‘S’ meant swimming, and ‘R’ meant running.  ‘B’ means basketball.  So this expression is asking for the intersection of all three sets – set ‘S’, set ‘R’ and set ‘B’.  In plain English it’s asking us for the students who do swimming and running and basketball.

Notice how now we have union symbols instead of intersection symbols.  This expression is asking us for all the students who belong to one or more of the three sets – ‘S’, ‘R’ and ‘B’.  In plain English, it’s asking us for any student who swims, runs or plays basketball, including students who do more than one activity.

We can get a mixture of union and intersection symbols as well, like this:

Now, brackets work like they do in any operation – we work out the things in the brackets first.  So we work on the ‘’ bit first.  This is a union symbol – so it’s saying all the students who are in either set ‘S’ or set ‘R’.  In plain English – all the students who do swimming or running.

Next part is the  bit.  This is an intersection symbol.  So if we put it together with the first bit, we’ve got:

=

(All the students who swim or run) and who play basketball

Notice how I’ve used the brackets in the final expression to emphasise how the sentence should be read.  The instructions that this mathematical expression is telling us are:

·         Find all the students who swim or run

·         From the group found in step 1, pick out the ones who also play basketball

·         The students picked in step 2 are your final set of students

### Combining in the other symbols

The ‘not’ symbol can also be used in these mathematical expressions of probability, like this for instance:

The expression inside the brackets is an intersection one – asking us for students who do both swimming and running.  But the ‘not’ symbol outside the brackets tells us to take all the students not in the set we’ve just found.  We’ve just found students who do both swimming and running.  So the students not in this set are any students who do not do both swimming and running, or do just basketball.  Using a step by step approach, it would look like:

·         Find the students who do both swimming and running

·         Find all the students who don’t belong to the set found in part 1

·         The students you found in step 2 are your final set of students

You might also get the ‘n’ and the ‘Pr’ symbols.  For instance:

The ‘n’ symbol makes this expression mean ‘the number of elements (students in this case) in the set .  This is the number of students who don’t do both swimming and running.

The ‘Pr’ in front of the  means the probability of picking a student from the class belonging to the set .  So, the probability of picking a student from the class who does not do both swimming and running.