Discrete Mathematics

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 is an exponential function. The base is b. (n times)

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 = bythen 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 If x is a positive integer, then b1/x = b For example, 91/2 = 9 = 3 x

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 x = x This is thelogarithm of x to the base b. Therefore we can conclude that logbbx = x. b

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: ai bk-i , where ai = 0 if i > n and bj = 0 if j > m. k i = 0

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) 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. xa

Limits • Alternative definition: • L is the limit of f(x) as x approaches a, and write: • L = lim f(x) 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. x a

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

Discrete Mathematics