MATHEMATICS IN NATURE

ANDREAL LIFE

Prepared by:Dr. Noha El-Attar

Faculty of Engineering

A PRETTY FACE?It is quite obvious that the

human face is symmetrical about a vertical axis down the nose.

However, studies have shown that the symmetry of another persons face is a large factor in determining whether or not we find them attractive.

In short, the better the symmetry of someone's face, the more attractive you should find them!

BEEHIVE BASICSA beehive is made up

of many hexagons packed together.

Why hexagons? Not squares or triangles?

Hexagons fit most closely together without any gaps, so they are an ideal shape to maximise the available space.

MATH IN CHEMISTRYExample: Suppose two sets of students

were measuring 36.0 mL of water. The following is their data:Group 1 = 13.6 Group 2= 0.5634.6 mL 35.9 mL34.2 mL 36.0 mL34.3mL 35.9 mL

Which group is accurate?

Percentage Error - used to determine how close to the true values, or how accurate, an experimental value actually is.

% Error = Experimental – Accepted Value x 100Accepted Value

For Group 1= 13.6% and Group 2= 0.56%

MATH IN PHARMACY

The strength or concentration of various drugs can be expressed as a ratio (fraction).

10 mg per ml = 10 mg/1 ml.

EXAMPLE 2 How many milliliters must be injected from

an ampule of a certain drug labeled "10 mg/2 ml" in order to administer a dose of 7.5 mg?

10 mg () = (7.5mg). (2 ml)

= 1.5 ml

SO……..We can find a strong relationship

between Mathematics and other Science.

That leads us to study more about it…

SETS AND SETS OPERATIONS

INTRODUCTION A set is a collection of objects having

specified characteristics.

The objects in a set are called elements of the set.

When talking about a set we usually denote the set with a capital letter (A, B, C …….Z).

When talking about the elements of the set, we usually a small letters (a,b,c,…….z).

HOW TO DESCRIBE A SET OF SOME ELEMENTS??

1- The Roster Notation:It defines the specific elements of the

set between brackets

Example:Let set A = The set of even numbers

greater than zero, and less than or equal10 of A={2, 4, 6, 8, 10}

2- EXTENSIONAL DEFINITION

Sometimes we can’t list all the elements of a set. For instance, Z = The set of integer numbers. We can’t write out all the integers, there infinitely many integers. So we adopt a convention using dots …

The dots mean continue on in this pattern forever and ever.

Z = { …-3, -2, -1, 0, 1, 2, 3, …}

3- SET – BUILDER NOTATION

When it is not convenient to list all the elements of a set, we use a notation the employs the rules in which an element is a member of the set. This is called set – builder notation.

V = { student | students who are registered in faculty of Pharmacy}

A = {x | x > 5} = This is the set A that has all real numbers greater than 5.

The symbol | is read as under condition.

NOTATION A well – defined set is a set in which we

know for sure if an element belongs to that set.

Example: The set of all students who are registered

in faculty of pharmacy is well – defined. (we can name every student “element” in our faculty “set”)

The set of all natural numbers which are less than 5: {0,1,2,3,4}.

The set of best TV shows of all time is not well – defined. (It is a matter of your opinion.)

SPECIAL SETS OF NUMBERS

N = The set of natural numbers. = {1, 2, 3, …}.

W = The set of whole natural numbers. ={0, 1, 2, 3, …} Z = The set of integers.

= { …, -3, -2, -1, 0, 1, 2, 3, …}

Rational Number: A number that can be made by dividing two integers.Q = The set of rational numbers. ={x| x=p/q, where p and q are elements of Z and q ≠ 0 but it can be 1}

EXAMPLES: 12/4= 3 3 Z 1/2 is a rational number (1 divided by 2) 0.75 (3/4) 1 (1/1) 2 (2/1) 2.12 (212/100) −6.6 (−66/10) 1.5 (3/2).

.

Irrational Number: is any real number that cannot be expressed as a ratio of integers.

Qc = The set of Irrational numbers. ={x| xp/q, where p and q are elements of Z}Examples: = ?/?

= THE SET OF REAL NUMBERS.= Q QC

N Z Q

which is the union of the set of rational numbers and the set of irrational numbers. 

REAL NUMBERS - CONT..

N Z

NZ

Q

QcR

C = THE SET OF COMPLEX NUMBERS.

A complex number is a number that can be expressed in the form a + bi.

It visually represented as a pair of numbers (a, b) forming a vector

on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the imaginary unit which satisfies i2 = −1.

THE MEMBERSHIP If A is a set and x is one of the objects of A,

this is denoted x ∈ A, and is read as "x belongs to A", or "x is an element of A". If y is not a member of A then this is written as y ∉ A, and is read as "y does not belong to B".

For example, with respect to the sets  A = {1,2,3,4}, B = {Black, white, red},

and F = {n2 − 4 | n is an integer; and 0 ≤ n ≤ 10} defined above,

4 ∈ A and 12 ∈ F; but 9 ∉ F and green ∉ B.

UNIVERSAL SET AND SUBSETS

The Universal Set denoted by U is the set of all possible elements used in a problem.

When every element of one set is also an element of another set, we say the first set is a subset.

A is a subset of B if and only if every x in A is belonging to B.

)( BxAxBA

Example:

if A={2, 3} and B={1,2,3,4,5}We say that A is a subset of B.

The notation we use is AB or BA, and BA.

THE EMPTY SET

The empty set is a special set. It contains no elements. It is usually denoted as { } or

The empty set is always considered a subset of any set.

Do not be confused by this question:

Is this set {0} empty? It is not empty! It contains the element zero.

CARDINAL NUMBER

The Cardinal Number of a set is the number of elements in the set and is denoted by n(A).

Let A={2,4,6,8,10}, then n(A)=5.

THE POWER SET It is a set which its all elements is a subset of A, and its

elements number is 2n , where n is the number of elements in the set A.

Let A={1,2,3}, list all the subsets of A. The subsets of S are:

A1={1}, A2={2}, A3= {3}, (three sets with one element)

A4= {1,2}, A5={1,3}, A6={2,3},(three sets with two elements)

A7={1,2,3}. (one set with tree elements)So the numbers of elements in the power set is equal ‘8’: 23 = {

SETS OPERATIONS1- INTERSECTION OF SETS

When an element of a set belongs to two or more sets we say the sets will intersect.

The intersection of a set A and a set B is denoted by A ∩ B.

A ∩ B = {x| x A and x B}

Example A={1, 3, 5, 7, 9} and B={1, 2, 3, 4, 5}

Then A ∩ B = {1, 3, 5}. Note that 1, 3, 5 are in both A and B.

FOR ANY THREE SETS A, B, C:

A A = A A U = A, A = A B = B A, (A B) C = A ( B C)

MUTUALLY EXCLUSIVE SETS

We say two sets A and B are mutually exclusive if A ∩ B = .

Think of this as two events that can not happen at the same time.

VENN DIAGRAMS FOR INTERSECTION

2- UNION OF SETS The union of two sets A, B is denoted by A U B. A U B = {x| x A or x B}

Example A={1, 3, 5, 7, 9} and B={1, 2, 3, 4, 5}

then the union of A, B is: A U B = {1, 2, 3, 4, 5, 7, 9}. The elements of the union are in A or in B

or in both. If elements are in both sets, we do not repeat them.

FOR ANY THREE SETS A, B, C: A A = A A U = U, A = A A B = A B, A B C = A ( B C)

VENN DIAGRAMS FOR UNION

DIFFERENCE BETWEEN SETS:

The difference between Set A and Set B is denoted by A-B.

A-B ={ x l xA and x B} Example: If A={1, 3, 5, 7, 9} and B={1, 2, 3, 4, 5}

then the difference between A, B is: A - B = {7, 9}.

VENN DIAGRAMS FOR DIFFERENCE

COMPLEMENT OF A SET

The complement of set A is denoted by A’, or by A

C. Ac = {x| x is not in set A}.

Example: U={1,2,3,4,5}, A={1,2},

then Ac = {3,4,5}.

AA C )(

,, CC ,)( CCC BABA

CCC AABA )(

i)

ii)

iii)

DE-MORGAN LOWS:

For every A, and B:

SPECIAL SYMBOLS: { } : Brackets- defining the set elements. | : under condition. = : equal : not equal : more than : less than : more than or equal : less than or equal : equal between sets “equivalent” : belong to : not belong to : Union : intersection : subset , , : not a subset