_{} 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:

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_{}

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.

What about something like this:

_{}

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.

John is just starting his first game of MegaRobotWars. He gets to jump into his own mechanoid robot machine and try and destroy 99 other robot warriors to win the game. He’s trying to choose what weapons and armour to mount on his robot, but this will depend on what the other robot warriors he’s facing have as equipment. 5 robots have only a) Draw a Venn diagram. Use ‘L’ for turbo lasers, ‘P’ for plasma rifles, and ‘A’ for armour. For the first robot John encounters calculate: b)
c)
d) Explain what the probability in part c) represents and how it might be significant |

Solution |

In this question we have three basic sets – robots which
have turbo lasers, robots which have plasma rifles and robots that have
armour. Now, most of the robots don’t have just one of these – most have at
least So to draw the Venn diagram, let’s start with three separate circles representing the three basic sets, and work out how they should go together later on: Stage 1 Now, as the diagram stands in stage 1, we have three different groups of robots, one group armed only with turbo lasers, one group armed only with plasma rifles and one with only armour. This doesn’t represent what we are trying to – lots of the robots have more than one type of equipment. The ‘super’ robots have all three pieces of equipment fitted. We can represent these by drawing the three circles so that they all overlap somewhat: Stage 2 The shaded area in the middle of the diagram represents
the intersection point of all three sets – so robots inside this shaded area
are armed with turbo lasers Now, we have actual numbers of how many robots there are equipped in each way, so we can label the diagram with numbers. For instance, only 5 robots have just turbo lasers, so we put a ‘5’ in the top left circle. The ‘super’ robots are represented by the central area – to work out how many of those there are we need to subtract all the other robot numbers from the total of 99. We can actually write this all in a mathematical form, using our set notation. In English, what we’re doing is:
Each term in this expression can be written using proper set and probability notation. For instance, the super robots are the robots that have everything. In other words, they are the robots at the intersection of all three sets. So the number of super robots can be written: The total number of robots is just the The number of robots that have Keep on doing this for the other groups and you should eventually get, as a mathematical replacement for the long word expression earlier:
Now, there is enough information in the question to substitute in for each of these terms: So there are 7 ‘super’ robots. This was a long winded way of going about it, but it contained lots of practice at converting everyday English expressions into mathematical notation. With this last piece of information, we can label every bit of the Venn diagram with a number of robots: A good check at this stage is to see whether all the separate areas of the Venn diagram add up to the total number of robots. In this case, 5 + 30 + 7 + 7 + 25 + 20 + 5 does add up to 99, so it’s OK. b)
c)
However, there’s this other Now all we have to do is add up the numbers inside this shaded area: So 42 robots fit into the set described by d) This is one of those horrible non-specific questions. The first part we can answer at least – the answer to part c) represents the probability of the first robot John encounters having lasers and/or plasma rifles, but no armour. But what about the significance of this? Well, these questions can be answered by thinking about
the |