We use
$n\left(S\right)$ to refer to the number of elements in a set
$S$ ,
$n\left(X\right)$ for the number of elements in
$X$ , etc.
In a box there are pieces of paper with the
numbers from 1 to 9 written on them. A piece of paper is drawn from the box andthe number on it is noted. Let
$S$ denote the sample space, let
$P$ denote the event 'drawing a prime number', and let
$E$ denote the event 'drawing an even
number'. Using appropriate notation, in how many ways is it possible to draw: i)any number? ii) a prime number? iii) an even number? iv) a number that is either
prime or even? v) a number that is both prime and even?
Drawing a prime number:
$P=\{2;3;5;7\}$
Drawing an even number:
$E=\{2;4;6;8\}$
The
union of
$P$ and
$E$ is the set of all elements in
$P$ or in
$E$ (or in both).
$P\cup E=2,3,4,5,6,7,8$ .
The
intersection of
$P$ and
$E$ is the set of all elements in both
$P$ and
$E$ .
$P\cap E=2$ .
100 people were surveyed to find out which fast food chain (Nandos, Debonairs or Steers) they preferred. The following results were obtained:
50 liked Nandos
66 liked Debonairs
40 liked Steers
27 liked Nandos and Debonairs but not Steers
13 liked Debonairs and Steers but not Nandos
4 liked all three
94 liked at least one
How many people did not like any of the fast food chains?
How many people liked Nandos and Steers, but not Debonairs?
The number of people who liked Nandos and Debonairs is 27, so this is the intersection of these two events. The number of people who liked Debonairs and Steers is 13, so the intersection of Debonairs and Steers is 13. We are told that 4 people like all three options, and so this means that there are 4 people in the intersection of all three options. So we can work out that the number of people who like just Debonairs is
$66\u20134\u201327-13=22$ (This is simply the total number who like Debonairs minus the number of people who like Debonairs and Steers, or Debonairs and Nandos or all three).
We draw the following diagram to represent the data:
We are told that there were 100 people and that 94 liked at least one. So the number of people that liked none is:
$100\u201394=6$ . This is the answer to a).
We can redraw the part of the Venn diagram that is of interest:
Total people who like Nandos: 50
Of these 27 like both Nandos and Debonairs, and 4 people like all three options. So we find that the total number of people who just like Nandos is:
$50\u201327\u20134=19$ Total people who like Steers: 40
Of these 13 like both Steers and Debonairs, and 4 like all three options. So we find that the total number of people who like just Steers is:
$40\u201313\u20134=23$ Now use the identity
$\mathrm{n(N\; or\; S)}=\mathrm{n(N)}+\mathrm{n(S)}\u2013\mathrm{n(N\; and\; S)}$ to find the number of people who like Nandos and Steers, but not Debonairs.
Which cellphone networks have you used or are you signed up for (e.g. Vodacom, Mtn or CellC)? Collect this information from your classmates as well. Then use the information to draw a Venn diagram (if you have more than three networks, then choose only the three most popular or draw Venn diagrams for all the combinations). Try to see if you can work out the number of people who use just one network, or the number of people who use all the networks.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?