Data Structures and Algorithms

1. Data Structures and Algorithms Discrete Math Review

2. Discrete Math review • Logarithmic Functions • Sets • Logic • Induction • Counting

3. Sets • A set is a collection of individual elements in the domain D. The universal set U contains every element in D . The null set  contains no element. • If A is a set in the domain D , A must be a subset of the universal set U , denoted as A  U. • If A consists of some but not all elements, A is then called a proper subset of U , denoted as A  U .

4. Sets • A set is a collection of definite and separate objects. • The cardinality of a set is the number of elements in the set. • A subset of a set is a set comprised of a sub-collection of the elements of the original set. • Example: Consider the set S = {2, 5, 7}. The subsets of S are the sets {2} , {5}, {7}, {2,5} , {2,7} , {5,7} , {2, 5, 7}and {} (the empty set). Fact: If a set has cardinality = n  , then the number of subsets = 2n .

5. Sets • A proper subset of a set is a set comprised of a sub-collection of some, but not all, of the elements of the original set. • Example: Consider the set S = {2, 5, 7}. The proper subsets of S are the sets {2} , {5}, {7}, {2,5} , {2,7} , {5,7} and {} (the empty set). • S = cardinality of 3  23  7?

6. Sets • There are only two different types of sets: • 1. An Infinite Set is a set that can be placed in a one-to-one correspondence to a proper subset of itself. •  The Basic Infinite Set is the set of positive whole numbers {1, 2, 3, 4, 5,……}. New Notation: The Basic Infinite Set has cardinality =   . • A Finite Set is a set that cannot be placed in a one-to-one correspondence to a proper subset of itself. • Something to think about….2 subsets ?

7. Logarithmic Functions • Logarithms were invented about 1614 by John Napier.  The word logarithm was derived from two Greek words, logos, which means "ratio," and arithmos, which means "number."  • Before calculators and computers were available, common logarithms were used to do certain kinds of calculations • Growth and decay • Big O notation growth of computational complexity

8. Logarithmic Functions • Logarithms are exponents • A logarithmic function is the inverse of an exponential function • One way to describe a logarithmic function is to interchange variables in the equation y = ax • Thus, x = ay is logarithmic

9. Logarithmic Functions • For logarithmic functions we use the notation logax • Therefore, The following are equivalent. • x = ay;  and • y = loga x

10. Logarithmic Functions Definition of Logarithm • Suppose b> 0 and .  For n > 0, there is a number p such that logbn = p if and only if bp = n. • log2256 = ?  2p = 256 • log101000 = ?  10p = 1000 • log72401 = ?  7p = 2401

11. Logic Examples: Boolean functions: NOT, AND, OR, XOR, . . . a NOT( a) 0 1 1 0

12. a NOT( a) 0 1 1 0 a b AND( a, b) 0 0 0 0 1 0 1 0 0 1 1 1 a b OR( a, b) 0 0 0 0 1 1 1 0 1 1 1 1 Logic Examples: Boolean functions: NOT, AND, OR

13. a b NOR( a, b) 0 0 1 0 1 0 1 0 0 1 1 0 a b XOR( a, b) 0 0 0 0 1 1 1 0 1 1 1 0 a b XNOR( a, b) 0 0 1 0 1 0 1 0 0 1 1 1 Logic Examples: Boolean functions: XOR, XNOR, NOR

14. A B F 0 1 0 1 0 1 1 1 1 Logic Boolean Simplification Express F in terms of A and B F = AB + AB Using the uniting theorem  A ( B + B ) = F

15. Series Arithmetic • One common type of series is the arithmetic series (also called an arithmetic progression). Each new term in an arithmetic series is the previous term plus a given number. For example this is an arithmetic series: 1+4+7+10+13+.... • In this case each term is the previous term plus 3. The difference between each term (the 3 in this case) is called the "common difference" and is generally denoted by the letter d. • There are many arithmetic series that have d=3. To specify which series we mean, we need to know one more piece of information: the value of the first term (usually called "a").

16. Series Arithmetic • If we're given a and d, then, that specifies a unique arithmetic series. All arithmetic series therefore have the following form: a+(a+d)+(a+2d)+(a+3d)+.... • So in general we can say that the value of the nth term is a+(n-1)d

17. Series Geometric • Another common type of series is the geometric series (also called a geometric progression). In this case, each term is the previous term multiplied by a given number. • Here's a geometric series, for example: 2+6+18+54+... • In this case, each term is the previous term multiplied by 3. The number you multiply by (in this case 3) is called the "common ratio" and is generally denoted by r. • There are many geometric series that have r=3. To specify which series we mean, again we need to know one more piece of information: the value of the first term (usually called "a").

18. Series Geometric • A geometric series is uniquely specified by the values of a and r. Every geometric series has the following form: a+ar+ar2+ar3+....... • Writing the series in this way we can see the formula for working out the value of any term. The first term is just a. The second term is a multiplied by r once. The third term is a multiplied by r twice, and so on. So the nth term must be a multiplied by r (n-1) times, that's arn-1.

19. Series Are there others?

20. The P word….Induction The Principle of Mathematical Induction • Suppose we have an assertion P(n) about the positive integers. • Then if we show both of (i) and (ii) below, then P(n) is true for all n >= 1. • (i). P(1) is true • (ii). For each k >= 1: If P(k) is true, then P(k+1) is true.

21. n(n+1) 2 The P word….Induction Prove that 1+2+3+…+ n = Prove that 1+3+5+…+ (2n – 1) = n2

22. Counting • Product Rule: n ways to do m tasks = nm ways. • How many different bit strings are there of length 7?

23. Counting • Pigenhole Principle: If N objects are placed into k boxes, then there is at least one box containing at least [N/k] objects. • Ten persons were born on the 1st, 11th and the 27th of the months of May, August and November. How many share the exact same birthday? Tells us nothing of the date. Asserts only that one exists!

24. n! P( n, r ) = (n – r)! Counting Permutations and Combinations • Permutations: Given that position is important, if one has 4 different objects (e.g. A, B, C and D), how many unique ways can they be placed in 3 positions (e.g. ABD, ACD, BDA, DBA, BAD, ADB, ADC, DCA, DAC, CDA etc.) • An ordered arrandement of r elements of a set is called a r-permutation.

25. n! C ( n, r ) = r! ( n-r )! Counting Permutations and Combinations • Combinations: If one has 4 different objects (e.g. A, B, C and D) how many ways can they be grouped as 2 objects when position does not matter (e.g. AB, AC, AD are correct but DA is not ok as is equal to AD.) • An r-combination is simply a subset of the set with r elements.

26. Base Conversion • The base value of a number system is the number of different values the set has before repeating itself. For example, decimal has a base of ten values, 0 to 9. • Binary = 2 (0, 1) • Octal = 8 (0 - 7) • Decimal = 10 (0 - 9) • Duodecimal = 12 (used for some purposes by the Romans) • Hexadecimal = 16 (0 - 9, A-F) • Vigesimal = 20 (used by the Mayans) • Sexagesimal = 60 (used by the Babylonians)

27. Base Conversion • Successive Division • 3710 base 6 • 37/6 = 6 r 1 • 6/6 = 1 r 0 • 1/6 = 0 r 1 • 3710 base 6 = 101

28. Base Conversion • 1016 base 10 • 1x60+0x61+1x62 = 1+0+36 = 37 • 5810base4 • 11110base13