## I.2 3. Some Elementary Logic, I.3 Some Fundamental Mathematical Definitions Pages 13 – 19

Some elementary logic: logical connectives, quantifiers,negation, free and bound variables and levels of formality followed by some fundamental mathematical definitions: the main number systems, the natural numbers, the integers, the rational numbers,the real numbers, the complex numbers and one of the four important mathematical structures: groups.

Mathematicians use two quantifiers “for all” denoted “ $\forall$“, and “there exists”, dentoted “\$“.  An example is given of the joke that makes use of the two different uses of the word “nothing”.

The logical connective between two mathematical sentences “and” is sometimes written “,“.  The mathematical “or”, “.“, includes the case when both mathematical sentences it ties are true.  A third connective is implies, “$\Rightarrow$“.  P $\Rightarrow$ Q means “if P then Q”, this statement is true under all circumstances except one, that is, if P is true and Q is false.  A cute example of this is provided where a young girl says to “put your hand up if you are a girl”.  Her brother puts his hand up on the grounds that this was not eliminated by the statement.  That is the girl did not add “and do not put your hand up if you are not a girl”.

The symbol “¬” is introduced to indicate “not”.  What is the negation of the sentence “Every number in the set A is odd”?  It is some member of A is even.

Free and bound variables are defined.   A variable is any letter used to stand for a mathematical object.  A free variable is free to take on any value.  A bound variable takes on a range of values specified by the mathematical notation.

There are various levels of formality used to communicate mathematical concepts.  Which one is used depends upon the audience.  An example is given as: “Every nonempty set of positive integers has a least element” can be completely formally written as:  $\forall$ A $\subset \!\,$ N  [(\$ n ÎN, n Î  A) $\Rightarrow$ ( x ÎA,  $\forall$ y  Î A, (y > x) . (y = x))], where A is a sub-set of positive integers and N the set of Natural numbers, “$\subset \!\,$” is defined to mean “subset”.

Numbers are a very basic mathematical concept.   More useful are collections of number with properties called number systems.

The natural numbers are the positive integers.  Sometimes zero is included and sometimes not.  One may add and multiply natural numbers but differences and division are not always possible.

The integers include the natural numbers, zero and the negative of the natural numbers.  The negative integer has the defining property that when you add the negative number to the integer the result is zero.  This sytem of numbers includes the addition, multiplication and subtraction properties.

The rational numbers, Q, include fractions or ratios of integers, except for not allowing for division by the zero element.  So these numbers have the addition, multiplication, and subtraction properties, as well as, division, except by zero properties.

The real numbers, denoted by R, are extensions of the rational numbers to include numbers such as the square root of 2 and pi.

The complex numbers, denoted by C, are a further extension of the real numbers to include for example, the square root of -1.  Inclusion of this one number and the real and imaginary parts of the complex number allows one to list roots of all polynomial equations as complex numbers.

Complex numbers may be writeen in polar coordinates.  Multiplication of complex numbers in polar coordinates is simply a contraction followed by a rotation of the point in two space or an Argand diagram.

There are four important algebraic strucures: groups, fields, vector spaces and rings.

Groups are important and occur when one considers symmetries.  Symmetries may arise as rigid motions of geometrical objects.  Rigid motions include translations and rotations but not stretching of any kind.  However, it should be noted that this is true in Euclidean space and not the actual space in which we live as well as for idealized objects and not those made up of actual matter.  Because the symmetries can be composed there are underlying groups of symmetires.  The composition operation is associative and has an identity element, it also includes and inverse.  Any set with a binary operation, such as composition of symmetries, that has these properties is called a group.  The composition does not have to be commutative.

If the group also has the commutative property then it is called Abelian.  Z, Q, R and C are all examples of Abelian groups under the operation of addition.