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.
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}.
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.