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Discrete Mathematics
Math Review
Math Review:
Exponents, logarithms, polynomials,
limits, floors and ceilings*
* This background review is useful for learning
how to analyze the time complexity of computer
algorithms.
Exponents

Let n be a positive integer, and b be a fixed
positive real number. Then the function
fb(n) = bn = b * b * b … * b
(n times)
is an exponential function. The base is b.
y = 5x
y = 2x
Exponentials with different bases
Rules for exponents
(a, b, x, and y assumed to be real numbers)
bxby = bx+y (not bxy !!)
b0 = 1
bx / by = bx-y
(bx )y = bxy
bx + bx = 2bx
For example, 2x + 2x = 2x+1
If bx = by then x = y.
Rules for exponents, continued
(a, b, x, and y assumed to be real numbers)
(ab)x = axbx
(a/b)x = ax/bx
If b is not equal to 0, then b–x = 1 / bx
x
1/x
If x is a positive integer, then b = b
For example, 91/2 = 9 = 3
Logarithms


Suppose b is a real number, with b >1, and x is
positive. Then fb(x) = bx is a strictly increasing
function of x, and it is a 1-1 correspondence.
Therefore it has an inverse, called the logarithmic
function to the base b (logbx).
This means:
blog bx = x
This is the logarithm of x to the base b.
Therefore we can conclude that logbbx = x.
Logarithms

Definition: bx = y if and only if logby = x

Example: 102 = 100 means log10100 = 2
Rules for logarithms
(b a real number greater than 1, and x and y positive real numbers)
logb(xy) = logbx + logby
logb(x/y) = logbx – logby
logb (x y)= y logbx
y = log2 x
Logarithmic function
Relationship between logarithms
with different bases

Theorem: logax = logbx / logba
Proof: Let X = logbx, Y = logba, and Z = logax.
By the definition of logarithm: bX = x, bY = a,
and aZ = x.
Thus bX = x = aZ = (bY)Z = bYZ
and therefore X = YZ
and therefore we conclude Z = X/Y.
Note on textbooks

When the textbooks refer to log x without
specifying a base, the base is assumed to be 2.
Factorial
n! = n (n-1)(n-2)(n-3)…1
 Example:
5! = 5*4*3*2*1
Polynomials


A polynomial is an expression of the form:
anx n + an-1xn-1 +… + a2x2 + a1x + a0
The ai are real numbers called coefficients, and variable x is
called an indeterminate. The largest exponent of the
indeterminate in the polynomial determines its order. The
order of the polynomial above is xn. A polynomial is
typically written in decreasing size of exponents.
Examples:


3x4 + 6x2 + x + 9
23x7 + 4x3 + 2
Rules for polynomials
Rule for addition of two polynomials:
(anxn + … + a2x2 + a1x + a0) +
(bnxn + … + b2x2 + b1x + b0) =
(an+bn)xn + … + (a2+b2)x2 + (a1+b1)x + (a0+b0)
Rule for multiplication of two polynomials:
(anxn + … + a2x2 + a1x + a0) *
(bmxm + … + b2x2 + b1x + b0) =
(anbm)xn+m + … + (a0 b2+ a1b1 + a2b0)x2 + (a0b1 + a1b0)x + (a0b0)


In general, for each k >= 0, the coefficient of xk in the product is:
k
i=0
ai bk-i , where ai = 0 if i > n and bj = 0 if j > m.
Intervals



An open interval (a,b) consists of all real numbers
between two fixed numbers a and b:
I = {x | a < x < b}
A closed interval [a,b] contains both endpoints:
I = {x | a <= x <= b}
A half-open interval (a,b] or [a,b) contains one endpoint:
I = {x | a < x <= b} or I = {x | a <= x < b}
Neighborhoods





The set of numbers that are close to a fixed number c is a
neighborhood of c. This implies that |x – c| is small.
A deleted neighborhood of c excludes c. In this case,
|x – c| > 0.
A symmetric neighborhood of c can be described by
|x – c| < h for some small positive number h.
A deleted symmetric neighborhood of c is described by
0 < |x – c| < h.
An open interval containing c is a neighborhood of c. For
example the open interval (c – h, c + h) is a symmetric
neighborhood of c.
Limits

Definition:
Suppose f is a function defined for values of x near a.
The domain of f need not include a, though it may. We
say that:
L is the limit of f(x) as x approaches a, and write:
L = lim f(x)
x a
provided that, for every real number h > 0 there is a
deleted neighborhood N of a such that:
L – h < f(x) < L + h
whenever x is in N and in the domain of f.
Limits




Alternative definition:
L is the limit of f(x) as x approaches a, and write:
L = lim f(x)
x a
provided that, for every real number h > 0 there exists a
real number d > 0 such that:
|f(x) – L| < h whenever 0 < |x – a| < d.
Translated to predicate logic:
h d x ((0 < |x – a| < d)
(|f(x) – L| < h))
when the universe of discourse for h and d is the set of
positive real numbers and for x is the set of real numbers.
Floors and Ceilings
For all real x and integer n:
x = the greatest integer less than or equal to x
x = the least integer less than or equal to x
n = n = n
x = n  n  x  n+1
x = n  x-1  n  x
x = n  n  x  n+1
x = n  x  n  x+1
x + n = x + n
n/2 + n/2 = n
x =  x
x =  x
Examples:
3.9 = 4
3.9 = 3
3.9 = 3 =  3.9
3.9 = 4 =  3.9