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Discrete Mathematics: Sets, Sequences and Functions

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  1. Discrete Mathematics:Sets, Sequences and Functions 1.1 Some Special Sets 1.2 Set Operations 1.3 Functions

  2. Discrete Mathematics • http://www.cs.tufts.edu/research/dmw/what_is_dm.html • http://en.wikipedia.org/wiki/Discrete_mathematics

  3. Sets • In the past few decades, it has become traditional to use set theory as the underlying basis for mathematics. • Set – a collection of objects • Must be unambiguous

  4. Sets • Sets – A, B, S, X • Objects – a, b, s, x • a is a member of S (a∈S) • a is not an element of S (a∉S)

  5. Some Special Sets • Natural numbers ℕ = {0, 1, 2, 3, …} • Positive integers ℙ = {1, 2, 3, …} • (some texts do not include 0 in ℕ) • Integers (Zahl) ℤ = {0, ±1, ±2, ±3, …} • Rational numbers (ratios of integers) ℚ = {m/n: m∈ℤ, n∈ℤ} • Real numbers ℝ

  6. Notation • Positive even numbers less than 12: {2, 4, 6, 8, 10} • Primes less than 20: {2, 3, 5, 7, 11, 13, 17, 19} • { : } • The colon is read “such that”

  7. Notation • {n:n∈ℕ and n is even} = {0, 2, 4, 6, …} • {x:x∈ℝ and 1≤x<3} • {n∈ℕ: n is even} • {x∈ℝ: 1≤x<3} • {n2: n∈ℕ} = {m∈ℕ: m = n2 for some n∈ℕ} = {0, 1, 4, 9, 16, …}= {n2: n∈ℤ} • {(-1)n: n∈ℕ} = {-1, 1}

  8. Definitions • Two sets are equal if they contain the same elements. • Order is irrelevant • No advantage or harm in repeating • {2, 4, 6, 8, 10} = {10, 8, 6, 4, 2} = {2, 8, 2, 6, 2, 10, 4, 2}

  9. Definitions • S is a subset of T (ST) if every element of S belongs to T • Thus, S = T iff ST and TS

  10. Examples • ℙℕ, ℕℤ, ℤℚ, ℚℝ • ℙℕℤℚℝ • {n∈ℙ: n is prime and n≥3}  {n∈ℙ: n is odd} • SS ( instead of )

  11. Notation • TS means TS and T≠S • T is a proper subset of S

  12. Interval Notation • [a, b] = {x∈ℝ: a≤x≤b} • [a, b) = {x∈ℝ: a≤x<b} • (a, b] = {x∈ℝ: a<x≤b} • (a, b) = {x∈ℝ: a<x<b} • [a, b] = closed interval • (a, b) = open interval • Intervals can also be used with ±∞

  13. Some Special Sets • {n∈ℕ: 2<n<3} • {x∈ℝ: x2<0} • {r∈ℚ: r2=2} • {x∈ℝ: x2+1=0} • Empty sets are denoted by { } and ∅ • Norwegian and Danish letter, not Greek Φ • ∅ is a subset of every set S, because x∈∅ implies x∈S

  14. Inception • Sets are objects, and can be members of other sets • Ex.: {{1, 2}, {1, 3}, {2}, {3}} has 4 members • Thus, {∅} has one member, and ∅ and {∅} are different. • ∅∈{∅}, and ∅{∅}, but ∅∉∅

  15. Power Sets • The set of all subsets of a set S is called the power set of S • P(∅) = {∅} • If S={a, b} and a≠b, then P(S)={∅, {a}, {b}, {a, b}} has 4 elements • If S = {a, b, c}, then P(S) = {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} has 8 elements • If S is a finite set with n elements, and if n≤3, then P(S) has 2n elements

  16. Languages • Alphabet = a finite nonempty set Σ whose members are symbols, or letters of Σ, and which is subject to some minor restrictions • Word = a finite string of letters from Σ • Σ* = the set of all words using letters from Σ • Language = any subset of Σ*

  17. Languages • Let Σ={a, b, c, ..., z} • Any string of letters from Σ belongs to Σ* • Σ* contains math, is, fun, aint, lieblich, amour, zzyzzoomph, etcetera, etc. (infinite) • We could define the American language L to be the subset of Σ* consisting of words in Webster’s New World Dictionary of the American Language. • Thus, L={a, aachen, aardvark, aardwolf,…, zymurgy} (finite)

  18. Null • The empty word, or null word, is the string with no letters at all, and is denoted by λ

  19. Restrictions on Σ • Σ cannot contain any letters that are themselves strings of letters in Σ • Σ={a, b, c} • Σ={a, b, c, ac} • Σ={a, b, ca} • Σ={a, b, Ab} • Σ={a, b, ac}

  20. Length • length(w) is the number of letters from Σ in w • length(aab); Σ={a, b} • length(bab); Σ={a, b} • length(abbAb); Σ={a, b, Ab} • length(λ)

  21. Set Operations • Union - A∪B = {x:x∈A or x∈B or both} • Intersection - A∩B = {x:x∈A and x∈B} • Disjoint – no elements in common (A∩B= ∅) • Relative complement – set of objects in A and not in B (A\B={x:x∈A and x∉B} = {x∈A:x∉B}

  22. Set Operations • Symmetric difference A⊕B={x:x∈A or x∈B but not both} • A⊕B=(A∪B)\(A∩B)=(A\B)∪(B\A) • Venn diagrams

  23. Universe • Set U is the universe or universal set • U can be ℕ, ℝ, or Σ* • Only consider elements in U and subsets of U • Absolute complement (or complement) of A, Ac=AU • U is denoted by a box in Venn diagrams • Note that A\B=A∩Bc • Ac∩Bc=(A∪B)c

  24. Commutative Laws • A∪B = B∪A • A∩B = B∩A

  25. Associative Laws • (A∪B)∪C = A∪(B∪C) • (A∩B)∩C = A∩(B∩C) • (A∪B)∩C ≟ A∪(B∩C) • (A∩B)∪C ≟ A∩(B∪C)

  26. Distributive Laws • (A∪B)∩C = (A∪B)∩(A∪C) • (A∩B)∪C = (A∩B)∪(A∩C)

  27. Idempotent Laws • A∪A = A • A∩A = A

  28. Identity Laws • A∪∅ = A • A∩U = U • A∩∅ = ∅ • A∪U = A

  29. Double Complementation • (Ac)c = A

  30. Other Laws • A∪Ac = U • A∩Ac = ∅ • Uc = ∅ • ∅c = U

  31. DeMorgan Laws • (A∪B)c = Ac∩Bc • (A∩B)c = Ac∪Bc

  32. Other Properties • (A∪B)∩Ac  B • (A⊕B)⊕C = A⊕(B⊕C)

  33. Product • Consider sets S and T, with s∈S and t∈T • (s, t) is an ordered pair (order is important) • Product – the set of all ordered pairs (s, t) • S x T = {(s, t): s∈S and t∈T} • If S = T, S x S can be written as S2

  34. Notation • For any finite set S, |S| indicates the number of elements in the set • |S x T| = |S| ∙ |T| • |P(S)| = 2|S| • P(S) can be written as 2S

  35. Product Set • Product set S1 x S2 x ∙∙∙ x Sn consists of all ordered n-tuples (s1, s2, …, sn)

  36. Functions • A function f assigns to some element x in some set S a unique element in a set T. • f is defined on S with values in T • S – domain of f, Dom(f) • The element assigned to x is written f(x)

  37. Functions • f is complete specified by Dom(f) and the formula or rule giving f(x) for each x∈Dom(f) • f(x) is the image of x under f • Im(f)T is the image of f, or the set of all images f(x)

  38. Functions • T is the codomain of f • Any set containing Im(f) can be a codomain • f:S→T means “f is a function with domain s and codomain T” • Or: f maps S into T

  39. Functions • Graph(f) = {(x, y)∈S x T: y = f(x)}