THE LANGUAGE OF MATHEMATICS
A set is a collection of objects. It can be described by the listing of its elements or an equation. B = {1, 2, 3, 4, 5, 6, 7, 8, 9} or B = {x/x is an integer and 0 < x < 10}. B is a set. A set with no elements is an empty set defined by f. f = {}. A universal set U is a set made of various sets which are subsets of U. Subsets are sets in which every element is an element of an other set. X = {a, b, c}; Z = {a, b, c, d} X is a subset of Z.
Any set A is a subset of itself, since any element in A is in A. If A is a subset of D and A does not equal D, then A is a proper subset of D. The set of all subsets of a set A, denoted P(A), is called the power set of A.
Union of two sets C and F, is the combination of all elements belonging to C or F.
C = {1, 2} and F = {a, b} Their union is C È F = {1, 2, a, b}
Intersection of two sets C and F, is the combination of all elements belonging to C and F. C = {1, 2, 3} and F = {2, 3, 4}. Their intersection C Ç F = {2, 3}.
Difference of two sets C and F, is the set made of all elements in C that are not in F.
C = {1, 3, 5} and F = {4, 5, 6}. Their difference is C – F = {1, 3}.
Imagine a universal set U and a subset N of U, the set U – N is called the complement of N and is written not N.
Venn diagrams provide pictorial views of sets. In a Venn diagram, a rectangle represents a universal set. Subsets of the universal set are drawn as circles. The inside of a circle represents the elements of that set.
A sequence is a list in which order is taken into account. 1, 2, 4, 8, 16, 32, 64,… is a sequence.
The index n of a sequence is the nth of the sequence.
A subsequence is a sequence resulting from an
original sequence. It retains only certain terms of the original and maintains
the order of terms.
A string is a finite sequence. Its length is the number of elements in the string.
The Binary number system represents integers using bits 0 and 1. The Hexadecimal number system represents integers using 16 symbols. The Decimal number system represents integers using the 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The following presentation can be used to convert a binary number to a decimal number:
101101 = 1 * 2^5 + 0*2^4 + 1*2^3 + 1*2^2 + 1*2^0 = 45 in decimal
Decimal numbers can be converted to binary by the computation of successive divisions as follow.
To convert the decimal number 16 in binary
2) 16 remainder = 0 1’s bit
2) 8 remainder = 0 2’s bit
2) 4 remainder = 0 4’s bit
2) 2 remainder = 0 8’s bit
2) 1 remainder = 1 16’s bit
16 = 10000 in binary
Hexadecimal numbers can be converted to decimal before being converted to binary.
Let S and T be two sets. Then a relation of S to T is a set of ordered couples (x,y) such that x є S
and y є T. Evidently a relation is any subset of the Cartesian product set S X T. A digraph is a pictorial way to represent a relation. In a digraph vertices are dots that represent the elements of the sets. A loop is used to join elements together in a digraph.
A relation R on a set X is reflexive when for each element x of X, (x, x) є R.
A relation R on a set X is symmetric when all x, y of X, (x, y) є R.
A relation R on a set X is antisymmetric when all x, y of X, (x, y) є R and x ≠ y, then (y, x) do not belong to R.
A relation R on a set X is transitive for all x, y, z of X, (x, y) and (y, z) є R, then (y, z) є Z.
When a relation is reflexive, transitive, and antisymmetric. It is called partial order.
An equivalence relation is a relation that is reflexive, symmetric, and transitive.
A matrix is a rectangular array of rows and columns of real numbers to represent a relation R from X to Y. The rows represent the elements of X, and the columns those of Y. Matrices can be added, subtracted, and multiplied to each other.
A database is a collection of records that are manipulated by computers. Database management systems are programs that allow users and administrators to access and manage the information in databases. They respond to queries which are requests for information from the database.
A function f is a special kind of relation from X to Y having the properties:
1. The domain of f is X
2. If (x, y), (x, y’) є f, then y = y’.
A function f from X to Y can be denoted f : X→ Y. The graph of a function f is another way to visualize that function. The graph is obtained by plotting points in the plane that correspond to the elements in f. The domain is contained in horizontal axis and the range is contained in the vertical axis.
1. Functions involving modulus operator techniques are used in programming languages such as
JavaScript. URL: http://msdn.microsoft.com/scripting/jscript/doc/jsoprMod.htm
2. Matrices are used in some calculator programs.
URL: ftp://www.maths.uq.edu.au/pub/krm/cmat/
QUESTIONS
1. Can 2 matrices of different shapes (# of rows and columns are different) be added?
The book has no mention of that.
2. Can anybody clarify the linear congruential method (2.8.14) for me?
I can’t figure out how this method is used to generate pseudorandom numbers.