Set Notation R.2
http://www.youtube.com/watch?v=7qD 1 womrlps
Pre-listening.
1. What is a set?
2. What are possible relations between two sets?
3. How do we denote sets of numbers?
Listening.
1) What does the speaker say about the relation between natural and whole numbers?
2) What does it mean when two sets are equal?...................
3) What is the difference between a subset and a proper subset?
4) How do we denote the empty set?......................................
5) What is interesting about the empty set?...............................
6) How do we denote a number which is not an element in some set?
7) Why is the example a true? .
8) Why is the example b false?
A graph enables us to represent the relation between two variables diagrummatically.
A setsquare enables us to draw or test a right angle. Now make similar sentences using the diagrams in exercise 1.
Section 2 Development
3. Read this:
Relations between sets
A set is any collection of things which we want to consider together. We use braces, {}, when we want to describe, or make a list of, the
elements of a set, and we use capital letters to denote sets. For example:
Let set A be {all animals) Let set G be {positive integers < 10}
Let set B be {lions} Let set H be {1, 2, 3,4, 5, 6, 7, 8, 9}
Let set C be {all animals except Let set I be {1,2,3}
Hons} Let set J be {0} (This set is called Let set D be {all human beings} the empty set)
Let set E be {all solid figures} Let set K be {3,4,5} Let set F be {all geometrical
figures}
4. Look and read:
Wc can use Venn diagrams or set notation to show the relations between sets. Look at the examples in this table:
Relation
; Set B is a proper subsci ofset
i A-
; Sei B is contained in set A.
i Set G is a subset ofset H and Í set H is ;i subset of sei G.
Í Set C is the complement of set
I B
' Set ! added to set K is the i union of I and K.
Venn diagram
Sei 1 subtracted from set K is the difference between K and I
The members common to set I and set K form the miersvciioii of K and I.
Set notation
G s H H c G
C= B'
Sets B and D are disjoint.      J £]Ť)
luK = {1,2,3.4,5}
K-I = {4,5} Kn 1= {3}
B n D = {0}
Now make a similar table to express the following relations:
a} A and D
b) EandF
c) H-I
d) Hul
e) Hnl
f) All triangles are plane figures.
g) No triangles are solid figures.
h) The set {figures with straight sides} and the set {figures with curved sides} have some members in common.
5. Using the following table, write definitions of the above relations between sets:
The union of two sets
The intersection of two sets
The difference between two sets
The complement of a set A
The subset A of a set B
The proper subset A of a set B
a set which contains all the members which do not belong to A. *■
a third set which contains all the members of both sets.
■a set in which every member of A is also in ft. but there is at least one member in B but not in A.
a third set which contains all the members of one set which are not common to both sets.
a third set which contains all the members common to both sets.
a set in which every member of A is also in B.
6. Read this:
The sign \j is used to symbolise union. A u B is read as 'A union B'. Now write similar sentences about the following signs:
a) r\
b) -
c)
d) <=
e) =>
f) >
g) ^
h) e
Section 3 Reading
7. Look and read:
■b) S
Fig. 5.1
Figure 5.1 shows a simple electric circuit with a power source, a switch and a light. In (a), the switch is open, so the light is off. In (b), the switch is closed, so the light is on.
32
Section 4 Listening
Sets of numbers
II. Listen to the passage and write down these words in the order in which you hear them:
common      frequently        referred to differs includes usually
enables        oblique whereas
12. Listen to the passage again and write down the symbols for the following sets as you hear them:
a) the empty set
b) the universal set
c) .all integers
d) all rational numbers ■ el all natural numbers I") all real numbers
13. Draw a Venn diagram representing the relations between the four sets (c), (d),
(c) and (f) in exercise 12. Using your diagram, say whether the following statements arc true or false. Correct the false statements.
a) NcR
b) N = R
c) NEZ* 6) NcZ*
e) Q => R
f) R.-Q = the set of all imaginary numbers
g) There are no disjoint sets in the diagram
Units
S^eSrSfdf numbers are used frequently in mathematics and the use of
standard abbreviations or symbols to refer to them enables us to save time and space. Capital letters are usually used for this notation. The set of natural numbers is denoted by N. 2 denotes the set of all integers. R represents all real numbers and Q all rational numbers.
Note that Z+ means all positive integers, R" all negative real numbers, and so on. Z+ differs from N in that Z+ consists of all positive integers, whereas N consists of all non-negative integers and therefore includes the element zero in addition to the elements of Z+. We can see from this that a set which contains only the element zero is not the same as the empty set, which contains no elements. Thus N-Z+ = {0}, but Z+ n Z~ = 0. ^J-jA^otHv^.
Some other sets are also referred to by abbreviations. We use a capital U to refer to the universal set, while the empty set is denoted by a symbol which consists of a zero bisected by an oblique line. This may also be read as the null or void set. For example, if two sets, A and B are disjoint (that is, they have no elements in common), then An B = 0..
ft 3®