211Definitions

Numbers

Natural numbers

Include 0,1,2,3,4,5...

Integers

Include 0,-1,1,-2,2,-3,3,-4,4...

Rational numbers

We say that a number is a rational number if it can be written as n/m, where n,m are integers and m != 0. In this case n is called nominator and m denominator of the rational number.

eg. 1, −5, 3/5 , −9/4...

Alphabets

A finite collection of symbols. e.g. {1, 2, d, r, $}. An alphabet with two symbols is called a binary alphabet. e.g. {a, b}

A word(string) is a sequence of symbols from an alphabet (say Sigma represents an alphabet).

Theorems

A statement that, objects satisfy certain properties. It must be true.

Methods of Proof

Dirrect proof method

If H then C. Solve by logically expanding on definitions.

Proof by cases method

This method can be explained as follows. Given a theorem of the form “If H then C”, the proof by cases method uses the following pattern. Usually the hypothesis H can be subdivided into several cases, say into three cases call them H1, H2 and H3. Then one needs to prove that each of these cases implies the conclusion C.

Proof by contradiction method

We assume that H is true but C is false. Then, using a sequence of logical reasoning we derive a contradiction such as 0 = 1.