Mathematical Preliminaries Chapter 2 Mathematical Preliminaries

Sets The term set is used to prefer to any collection of objects, which are called members or elements of the set. A set is called finite if it contains n elements, for some constant n0, and infinite otherwise. An infinite set is called countable if its elements can be listed as the first element, second element, and so on; otherwise it is called uncountable. Cardinality, subset, union, intersection, difference, complement, disjoint, power set

Relations The Cartesian product: AB={(a,b)|aA, bB} A binary relation, or simply a relation, R from A to B is a set of ordered pairs (a,b) where aA and bB, that is, RAB. If A=B, R is a relation on the set A. Domain, range, symmetric/asymmetric/antisymmetric, reflexive/irreflexive, transitive, partial order A relation R on a set A is called an equivalence relation if it is reflexive, symmetric and transitive. In this case, R partitions A into equivalence classes C1, C2,..., Ck such that any two elements in one equivalence class are related by R.

Functions A function f is a (binary) relation such that for every element xDom(f) there is exactly one element yRan(f) with (x,y)f. In this case one usually writes f(x)=y instead of (x,y)f and says that y is the value or image of f at x. one to one, onto, bijection, one to one correspondence

Direct proof To prove that “PQ”, a direct proof works by assuming that P is true and then deducing the truth Q from the truth of P.

Indirect proof The implication “PQ” is logically equivalent to the contrapositive implication “QP”.

Proof by contradiction To prove that the statement “PQ” is true, we start by assuming that P is true but Q is false. If this assumption leads to a contradiction, it means that the assumption that “Q is false” must be wrong, and hence Q must follow from P.

Proof by counterexample When we are faced with a problem that requires proving or disproving a given assertion, we may start by trying to disprove the assertion with a counterexample. It is usually employed to show that a proposition that holds true quite often is not always true.

Mathematical induction First we prove that the property holds for n0. This is called the basis step. Then we prove that whenever the property is true for n0, n0+1, ... , n-1, then it must follow that the property is true for n. This is called the induction step. We then conclude that the property holds for all values of nn0.

A Puzzle for Pirates Ten pirates have got their hands on a hoard of 100 goldpieces, and wish to divide the loot between them. They are democratic pirates, in their own way, and it is their custom to make such divisions in the following manner. The fiercest pirate makes a proposal about the division, and everybody votes --- one vote each including the proposer. If 50% or more are in favour, the proposal passes and is implemented forthwith. Otherwise the proposer is thrown overboard and the procedure is repeated with the next fiercest pirate. All the pirates enjoy throwing people overboard, but given the choice they prefer hard cash. They dislike being thrown overboard themselves. All pirates are rational, know that the other pirates are rational, know that they know that... and so on. Moreover, no two pirates are equally fierce, so there is a precise 'pecking order' --- and it is known to them all. Finally: gold pieces are indivisible and arrangements to share pieces are not permitted (since no pirate trusts his fellows to stick to such an arrangement). It's every man for himself. Which proposal will maximize the fiercest pirate's gain? If the number of pirates is 20, 50,100,200,400,…, what is the result?

Logarithms

Floor and ceiling functions Let x be a real number. The floor of x, denoted by x, is defined as the greatest integer less than or equal to x. The ceiling of x, denoted by x, is defined as the least integer greater than or equal to x. x/2+x/2=x -x=-x, -x=-x Theorem: Let f(x) be a monotonically increasing function such that if f(x) is integer, then x is integer. Then f(x)=f(x) and f(x)=f(x)

Factorials 0!=1 n!=n(n-1)! if n1 Stirling’s formula:

Binomial coefficient

The pigeonhole principle Theorem 2.3 If n balls are distributed into m boxes, then 1) one box must contain at least n/m balls 2) one box must contain at most n/m balls

Summations

Approximation of summations by integration Let f(x) be a continuous function that is monotically decreasing Let f(x) be a continuous function that is monotically increasing

Recurrence relations A recursive formula is simply a formula that is defined in terms of itself. A recurrence relation is called linear homogeneous with constant coefficients if it is of the form f(n)=a1f(n-1)+a2f(n-2)+...+akf(n-k) In this case, f(n) is said to be of degree k. If an additional term involving a constant or a function of n appears in the recurrence, then it is called inhomogeneous.

Solution of recurrence relation Solution of linear homogeneous recurrences: characteristic equation Solution of inhomogeneous recurrences: some elementary inhomogeneous recurrences Solution of divide-and-conquer recurrences: 1) expanding the recurrence/Master Theorem 2) substitution 3) change of variables Some important results!

A Important Theorem Theorem 2.5: Let a and c be nonnegative integers, b, d and x nonnegative constants, and let n=ck, for some nonnegative integer k. Then the solution to the recurrence f(n)=af(n/c)+bnx is 1) If a<cx, then f(n)=(nx) 2) If a=cx, then f(n)=(nxlogn) 3) If a>cx, then f(n)=(n^logca)

Master Theorem Let a>=1, c>1 be constants, f(n) be a function, T(n) be defined on the negative integers by the recurrence T(n)=aT(n/c)+f(n) where we intepret n/c to mean either n/c or n/c. Then 1) If f(n)=O(n^(logca-)) for some >0, then T(n)=(n^logca) 2) If f(n)=(n^logca), then T(n)=((n^logca)logn) 3) If f(n)=(n^(logca+)) for some >0 and if a*f(n/c)kf(n) for some constant k<1 and all sufficiently large n, then T(n)=(f(n))