MATHEMATICS EXPLORER
Why Mathematics?
It is important to note that mathematics graduates are
generally highly respected and valued by potential employers and this is worth
considering. A good knowledge of mathematics opens up many doors for you as a
person. Those students who take mathematics seriously and do well in it end up
with good disciplines, good careers, good pay and steady jobs. Experience and
experiment have also shown that mathematicians and those good in mathematics
are better managers, better accountant and coordinators of men and resources.
Without mathematics you cannot be an engineer, an accountant, a banker, a
computer scientist, a medical doctor and you cannot read a good degree in the
University. Take mathematics very seriously and have plenty of fun with it.
This month’s exploration:
Sets Theory
1. Sets and
Elements
A collection of objects
is called a set. The individual objects in a set are called the elements
of the set. For example, Sunday, Monday, Tuesday, Wednesday,
Thursday, Friday, and Saturday are the elements of the set
of the days of the week.
2. Symbols for Sets and their Elements
Sets may be represented
by capital letters, such as A, B, C, etc. The elements of a set may be
represented by small letters, such as a, b, c, etc. Then the statements a is
an element of the set A, or a belongs to A, or a is in A all
have the same meaning, and are written in symbols as follows:
a is in A.
For example we may let A stand for the set of
days of the week, and we may let a = Sunday, b = Monday, c = Tuesday, etc. Then
a is in A means Sunday is a day of the week, b is in A means Monday
is a day of the week, etc.
EXERCISE
Let A stand for the set
of months, and let a = January, b = February, etc. Give the meaning of the
symbols: a is in A, b is in A, etc.
3. Sets
Defined by Lists of Elements
A set may be defined by
listing its elements in brackets. For example, if A stands for the set of
colours used in traffic lights, then we may write:
A = {red, yellow, green}.
EXERCISE
Give a definition of
the set of days in the week by listing its elements in brackets.
4. Sets Defined by Properties of Elements
A set may be defined by
means of a variable x and an incomplete statement P(x) giving the
characteristic property of the elements in the set. The statement A is the
set of all elements x such that P(x) is written in symbols as follows:
A = {x: P(x)}.
For example, let x stand for a person, and let
P(x) = x is an artist. Then {x: P(x)} is the set of all artists.
EXERCISE
Let x stand for an
animal, and let P(x) = x has four legs. Say which of the following animals
belong to the set {x: P(x)}:
- A
buffalo
- A cat
- A
monkey
- A
snake.
5.
Equality of Sets
Two sets are called equal
when they have exactly the same elements. When a set is defined by listing its
elements the list may be in any order. For example, the sets of colours
{yellow, red, green}, {green, yellow, red} and {red, yellow, green} are all
equal.
If the same set is defined by different
incomplete statements, then the incomplete statements must have the same
meaning. For example, let x stand for a day of the week. Then we have
{Saturday, Sunday} = {x: x is at the weekend} =
{x: x is the day after Friday or the day before Monday}.
EXERCISE
Let x stand for a whole
number. Say which of the following pairs of sets are equal and which are not
equal.
- {1, 2,
3, 4, 5}
{x: x is odd and less than 6} - {1,
3, 5}
{5, 3, 1} - {x:
x is positive and not greater than 5}
{x: 0 < x < 6}.
6. Union of Sets
From two given sets A
and B we can make a new set that consists of all the elements of A and all the
elements of B. This new set is called the union of A and B. It is
represented by the symbol A + B.
For example, let A = {a, b, c, d} and let B =
{c, d, e}. Then A + B = {a, b, c, d, e}. Notice that the elements c and d are
in A as well as B, but they are written only once in the list for A + B.
The union of two sets is defined in symbols as
follows:
A + B = {x: x is in
A or x is in B}.
EXERCISE
Give the unions of the
following pairs of sets.
- A =
{Friday, Saturday, Sunday, Monday}
B = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}. - A =
{x: x is a man over 40 years old}
B = {x: x is a woman over 40 years old}.
7. Commutative Law for Unions
In the union of two
sets it does not matter which set is written first. For example,
{a, b, c} + {c, d} = {a, b, c, d}
and
{c, d} + {a, b, c} = {a, b, c, d}.
This may also be seen in the definitions
A + B = {x: x is in A or x is in B}
and
B + A = {x: x is in B or x is in A}
because, by the commutative law for the logical
OR, x is in A or x is in B has the same meaning as x is in B or x is
in A.
This fact is called the commutative law for the
union of sets. It is summarized in symbols as follows:
A + B = B + A.
8. Associative Law for Unions
The union of A + B with
another set C is a composite union:
(A + B) + C = {x: (x is in A or x is in B) or x
is in C}.
By the associative law for the logical OR, this
is equal to:
{x: x is in A or (x is in B or x is in C)} = A +
(B + C).
Therefore we have the associative law for
unions, which is summarized as follows:
(A + B) + C = A + (B
+ C).
As a result of this law
we may omit the brackets and write:
A + B + C.
For example, let A be the set of all African
people, let B be the set of all Asian people, and let C be the set of all
European people. Then the union A + B + C is the set of all the people in
Africa, Asia, and Europe.
9. Intersection of Sets
From two given sets A
and B we can make a new set that consists of all the elements that belong to
both A and B at the same time. This new set is called the intersection
of A and B. It is represented by the symbol A . B.
For example, let A = {a, b, c, d} and let B =
{c, d, e}. Then A . B = {c, d}.
The intersection of two sets is defined in
symbols as follows:
A . B = {x: x is in
A and x is in B}.
EXERCISE
Give the intersections
of the following pairs of sets.
- A =
{2, 4, 6, 8, 10, 12}
B = {3, 6, 9, 12}. - A =
the time from 6:00 a.m. to 6:00 p.m.
B = the time from 12:00 noon to 12:00 midnight.
10. Commutative Law for Intersections
In the intersection of
two sets it does not matter which set is written first. For example,
{a, b, c, d} . {c, d, e} = {c, d}
and
{c, d, e} . {a, b, c, d} = {c, d}.
This may also be seen in the definitions
A . B = {x: x is in A and x is in B}
and
B . A = {x: x is in B and x is in A}
because, by the commutative law for the logical
AND, x is in A and x is in B has the same meaning as x is in B and x
is in A.
This fact is called the commutative law for the
intersection of two sets. It is summarized in symbols as follows:
A . B = B . A.
11. Associative Law for Intersections
The intersection of A .
B with another set C is a composite intersection:
(A . B) . C = {x: (x is in A and x is in B) and
x is in C}.
By the associative law for the logical AND, this
is equal to
{x: x is in A and (x is in B and x is in C)} = A
. (B . C).
Therefore we have the associative law for
intersections, which is summarized in symbols as follows:
(A . B) . C = A . (B
. C).
As a result of this law
we may omit the brackets and write
A . B . C.
For example, let A be the set of all African
people, let B be the set of all female people, and let C be the set of all
children under 10 years old. Then A . B . C is the set of all African girls
under 10 years old.
12. Distributive Law: Union
over Intersection
The union of a set A
with an intersection B . C is a composite expression:
A + (B . C) = {x: x is in A or (x is in B and x
is in C)}.
By one of the distributive laws for composite
statements, x is in A or (x is in B and x is in C) has the same meaning
as (x is in A or x is in B) and (x is in A or x is in C). Therefore
A + (B . C) = {x: (x is in A or x is in B) and
(x is in A or x is in C)}.
But
{x: (x is in A or x is in B) and (x is in A or x
is in C)} = (A + B) . (A + C).
Therefore
A + (B . C) = (A +
B) . (A + C).
This is one of the
distributive laws for sets.
For example, suppose that applicants for a job
must have either a university degree, or five years of work experience and a
certificate of English language ability. Let A be the set of people with a
degree, let B be the set of people with five years of experience, and let C be
the set of people with a certificate of English. Then A + (B . C) is the set of
people who may apply for the job. By the distributive law this is the same as
(A + B) . (A + C). This shows that the applications may be checked in two
separate ways before being accepted. One check makes sure that the applicant
belongs to A + B (has a degree or five years of experience). The other check
makes sure that the applicant belongs to A + C (has a degree or a certificate
of English).
EXERCISE
Suppose a person has
enough money to buy a small new motor-car or a large second-hand motor-car. Let
A be the set of small cars, let B be the set of large cars, and let C be the
set of second-hand cars. Use the distributive law to find another way of saying
that this person may buy a car which is small or large, and small or
second-hand.
13. Distributive Law: Intersection over Union
The intersection of a
set A with a union B + C is a composite expression:
A . (B + C) = {x: x is in A and (x is in B or x
is in C)}.
By one of the distributive laws for composite
statements, x is in A and (x is in B or x is in C) has the same meaning
as (x is in A and x is in B) or (x is in A and x is in C). Therefore
A . (B + C) = {x: (x is in A and x is in B) or
(x is in A and x is in C)}.
But
{x: (x is in A and x is in B) or (x is in A and
x is in C)} = (A . B) + (A . C).
Therefore
A . (B + C) = (A .
B) + (A . C).
This is another
distributive law for sets.
For example, let A be the set of children from 5
to 12 years old, let B be the set of boys, and let C be the set of girls. Then
A . (B + C) means: the set of children who are 5 to 12 years old and either
boys or girls. This is the same as the set (A . B) + (A . C), which means:
children who are boys 5 to 12 years old or girls 5 to 12 years old.
EXERCISE
Let A be the set of
windy days in a particular year at a particular place, let B be the set of
sunny days, and let C be the set of rainy days. Give the meaning of the two
composite expressions:
- A .
(B + C)
- (A .
B) + (A . C).
14. Universal Set
The universal set
contains all the individual objects being studied.
For example, if the sets being studied consist
of men, women, boys, and girls in a population, then the universal set is
everyone in the population.
We shall use the letter U to stand for the
universal set.
EXERCISE
What is the universal
set if the sets being studied consist of elements such as the following: 10
January, 5 February, 15 March, 1 May, 28 June, 16 November, 23 December?
15. Empty Set
The empty set
does not contain anything. In other words, it is the set that has no elements.
We shall use the letter O to stand for the empty set.
For example, let A be the set of all men, and
let B be the set of all women. Then the intersection of these two sets is
empty, that is: A . B = O.
EXERCISE
Say whether or not the
intersection of the following pairs of sets is the empty set.
- {1,
2, 3}
{2, 4, 6} - {a,
b, c}
{d, e, f}.
To be continued.
This article and work is by Professor R. H. B. Exell, M.A., D. Phil. (Oxford); Hon. D. Sc. (KMUTT). Joint Graduate
School of Energy and Environment, King
Mongkut's University of Technology Thonburi, Bangmod, Thungkru, Bangkok 10140,
Thailand.
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