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Chapter 2: Basic Structures: Sets, Functions, Sequences and Sums

Chapter 2: Basic Structures: Sets, Functions, Sequences and Sums. Sets (Section 2.1) Set Operations (Section 2.2) Functions (Section 2.3) Sequences and Summations (Section 2.4). Sets (2.1). A set is a collection or group of objects or elements or members . (Cantor 1895)

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Chapter 2: Basic Structures: Sets, Functions, Sequences and Sums

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  1. Chapter 2:Basic Structures: Sets, Functions, Sequences and Sums Sets (Section 2.1) Set Operations (Section 2.2) Functions(Section 2.3) Sequences and Summations (Section 2.4)

  2. Sets (2.1) • A set is a collection or group of objects or elements or members. (Cantor 1895) • A set is said to contain its elements. • There must be an underlying universal set U, either specifically stated or understood. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  3. Sets (2.1) (cont.) • Notation: • list the elements between braces: S = {a, b, c, d}={b, c, a, d, d} (Note: listing an object more than once does not change the set. Ordering means nothing.) • specification by predicates: S= {x| P(x)}, S contains all the elements from U which make the predicate P true. • brace notation with ellipses: S = { . . . , -3, -2, -1}, the negative integers. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  4. Sets (2.1) (cont.) • Common Universal Sets • R = reals • N = natural numbers = {0,1, 2, 3, . . . }, the counting numbers • Z = all integers = {. . , -3, -2, -1, 0, 1, 2, 3, 4, . .} • Z+ is the set of positive integers • Notation: x is a member of S or x is an element of S: x  S. x is not an element of S: x  S. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  5. Sets (2.1) (cont.) • Subsets • Definition: The set A is a subset of the set B, denoted A  B, iff x [x  A  x  B] • Definition: The void set, the null set, the empty set, denoted , is the set with no members. Note: the assertion x  is always false. Hence x [x  x  B] is always true(vacuously). Therefore,  is a subset of every set. Note: A set B is always a subset of itself. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  6. Sets (2.1) (cont.) • Definition: If A  B but A  B the we say A is a proper subset of B, denoted A  B (in some texts). • Definition: The set of all subset of a set A, denoted P(A), is called the power set of A. • Example: If A = {a, b} then P(A) = {, {a}, {b}, {a,b}} CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  7. Sets (2.1) (cont.) • Definition: The number of (distinct) elements in A, denoted |A|, is called the cardinality of A. If the cardinality is a natural number (in N), then the set is called finite, else infinite. • Example: A = {a, b}, |{a, b}| = 2, |P({a, b})| = 4. A is finite and so is P(A). Useful Fact: |A|=n implies |P(A)| = 2n CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  8. Sets (2.1) (cont.) • N is infinite since |N| is not a natural number. It is called a transfinite cardinal number. • Note: Sets can be both members and subsets of other sets. • Example: A = {,{}}. A has two elements and hence four subsets: , {}, {{}}. {,{}} Note that  is both a member of A and a subset of A! • Russell's paradox: Let S be the set of all sets which are not members of themselves. Is S a member of itself? • Another paradox: Henry is a barber who shaves all people who do not shave themselves. Does Henry shave himself? CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  9. Sets (2.1) (cont.) • Definition: The Cartesian product of A with B, denoted A x B, is the set of ordered pairs {<a, b> | a  A  b  B} Notation: Note: The Cartesian product of anything with  is . (why?) • Example: A = {a,b}, B = {1, 2, 3} AxB = {<a, 1>, <a, 2>, <a, 3>, <b, 1>, <b, 2>, <b, 3>} What is BxA? AxBxA? • If |A| = m and |B| = n, what is |AxB|? CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  10. Set Operations (2.2) (cont.) • Propositional calculus and set theory are both instances of an algebraic system called a Boolean Algebra. The operators in set theory are defined in terms of the corresponding operator in propositional calculus As always there must be a universe U. All sets are assumed to be subsets of U CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  11. Set Operations (2.2) (cont.) • Definition:Two sets A and B are equal, denoted A = B, iff x [x  A  x  B]. • Note: By a previous logical equivalence we have A = B iff x [(x  A  x  B)  (x  B  x  A)] or A = B iff A  B and B  A CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  12. Set Operations (2.2) (cont.) • Definitions: • The union of A and B, denoted A U B, is the set {x | x  A  x  B} • The intersection of A and B, denoted A  B, is the set {x | x  A  x  B} Note: If the intersection is void, A and B are said to be disjoint. • The complement of A, denoted , is the set {x | (x  A)} Note: Alternative notation is Ac, and {x|x  A}. • The difference of A and B, or the complement of B relative to A, denoted A - B, is the set A  Note: The (absolute) complement of A is U - A. • The symmetric difference of A and B, denoted A  B, is the set (A - B) U (B - A)

  13. Set Operations (2.2) (cont.) • Examples:U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A= {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}. Then • AB = {1, 2, 3, 4, 5, 6, 7, 8} • A  B = {4, 5} • = {0, 6, 7, 8, 9, 10} • = {0, 1, 2, 3, 9, 10} • A - B = {1, 2, 3} • B - A = {6, 7, 8} • AB = {1, 2, 3, 6, 7, 8} CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  14. Set Operations (2.2) (cont.) • Venn Diagrams • A useful geometric visualization tool (for 3 or less sets) • The Universe U is the rectangular box • Each set is represented by a circle and its interior • All possible combinations of the sets must be represented • Shade the appropriate region to represent the given set operation. U U B A A B C For 3 sets For 2 sets CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  15. Set Operations (2.2) (cont.) • Set Identities • Set identities correspond to the logical equivalences. • Example: The complement of the union is the intersection of the complements: =  Proof: To show: x [x   x   ] To show two sets are equal we show for all x that x is a member of one set if and only if it is a member of the other. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  16. Set Operations (2.2) (cont.) • We now apply an important rule of inference (defined later) called Universal Instantiation In a proof we can eliminate the universal quantifier which binds a variable if we do not assume anything about the variable other than it is an arbitrary member of the Universe. We can then treat the resulting predicate as a proposition. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  17. We say 'Let x be arbitrary.' Then we can treat the predicates as propositions: CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  18. Set Operations (2.2) (cont.) Hence x  x  is a tautology. Since • x was arbitrary • we have used only logically equivalent assertions and definitions CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  19. Set Operations (2.2) (cont.) we can apply another rule of inference called Universal Generalization We can apply a universal quantifier to bind a variable if we have shown the predicate to be true for all values of the variable in the Universe. and claim the assertion is true for all x, i.e., x [x  x  ] Q. E. D. (Latin phrase “Quod Erat Demonstrandum”) CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  20. Set Operations (2.2) (cont.) • Note: As an alternative which might be easier in some cases, use the identity A = B  [A  B and B  A] • Example: Show A  (B - A) =  The void set is a subset of every set. Hence, A  (B - A)  Therefore, it suffices to show A  (B - A)  or x [xA  (B - A)  x ] So as before we say 'let x be arbitrary’. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  21. Set Operations (2.2) (cont.) • Example (cont.) Show xA  (B - A)  x is a tautology. But the consequent is always false. Therefore, the antecedent better always be false also. Apply the definitions: CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  22. Set Operations (2.2) (cont.) • Example (cont.) Hence, because P  P is always false, the implication is a tautology. The result follows by Universal Generalization. Q. E. D. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  23. Set Operations (2.2) (cont.) • Union and Intersection of Indexed Collections • Let A1,A2 ,..., An be an indexed collection of sets. • Union and intersection are associative (because 'and' and 'or' are) we have: CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  24. Set Operations (2.2) (cont.) • ExamplesLet CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  25. Functions (2.3) • Definition:Let A and B be sets. A function (mapping, map) f from A to B, denoted f :AB, is a subset of A*B such that x [x  A  y [y  B  < x, y > f ]] and [< x, y1 > f  < x, y2 >  f ]  y1 = y2 CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  26. Functions (2.3) (cont.) • Note: f associates with each x in A one and only one y in B. A is called the domain and B is called the codomain. If f(x) = y y is called the image of x under f x is called a preimage of y (note there may be more than one preimage of y but there is only one image of x). The range of f is the set of all images of points in A under f. We denote it by f(A). CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  27. Functions (2.3) (cont.) If S is a subset of A then f(S) = {f(s) | s in S}. Example: • f(a) = Z • the image of d is Z • the domain of f is A = {a, b, c, d} • the codomain is B = {X, Y, Z} • f(A) = {Y, Z} • the preimage of Y is b • the preimages of Z are a, c and d • f({c,d}) = {Z} A B a X b Y c Z d CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  28. Functions (2.3) (cont.) • Injections, Surjections and Bijections • Let f be a function from A to B. • Definition: f is one-to-one (denoted 1-1) or injective if preimages are unique. Note: this means that if a  b then f(a)  f(b). • Definition: f is onto or surjective if every y in B has a preimage. Note: this means that for every y in B there must be an x in A such that f(x) = y. • Definition: f is bijective if it is surjective and injective (one-to-one and onto). CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  29. Functions (2.3) (cont.) • Examples: The previous Example function is neither an injection nor a surjection. Hence it is not a bijection. A B a X b Y c Z d Surjection but not an injection CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  30. Functions (2.3) (cont.) A B A B a a V V b b W W c c X X d d Y Y Injection & a surjection, hence a bijection Z Injection but not a surjection CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  31. Functions (2.3) (cont.) • Note: Whenever there is a bijection from A to B, the two sets must have the same number of elements or the same cardinality. • That will become our definition, especially for infinite sets. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  32. Functions (2.3) (cont.) • Examples: Let A = B = R, the reals. Determine which are injections, surjections, bijections: • f(x) = x, • f(x) = x2, • f(x) = x3, • f(x) = x + sin(x), • f(x) = | x | CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  33. Functions (2.3) (cont.) • Let E be the set of even integers {0, 2, 4, 6, . . . .}. Then there is a bijection f from N to E , the even nonnegative integers, defined by f(x) = 2x. Hence, the set of even integers has the same cardinality as the set of natural numbers. OH, NO! IT CAN’T BE....E IS ONLY HALF AS BIG!!! Sorry! It gets worse before it gets better. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  34. Functions (2.3) (cont.) • Inverse Functions • Definition: Let f be a bijection from A to B. Then the inverse of f, denoted f-1, is the function from B to A defined as f-1(y) = x iff f(x) = y CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  35. Functions (2.3) (cont.) • Example: Let f be defined by the diagram: A f-1 B A f B a a V V b b W W c c X X d d Y Y Note: No inverse exists unless f is a bijection CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  36. Functions (2.3) (cont.) • Definition: Let S be a subset of B. Then f-1(S) = {x | f(x)  S} Note: f need not be a bijection for this definition to hold. • Example: Let f be the following function: A B a X f-1({Z}) = {c, d} f-1({X, Y}) = {a, b} b Y c Z d CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  37. Functions (2.3) (cont.) • Composition • Definition:Let f: B C, g: A B. The composition of f with g, denoted fg, is the function from A to C defined by f  g(x) = f(g(x)) CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  38. A g B f C a V h • Examples: b W i c X j d Y A fg C a h b i c j d CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  39. Functions (2.3) (cont.) • If f(x) = x2 and g(x) = 2x + 1, then f(g(x)) = (2x+1)2 and g(f(x)) = 2x2 + 1 • Definition: • The floor function, denoted f ( x) = x or f(x) = floor(x), is the largest integer less than or equal to x. • The ceiling function, denoted f ( x) = x or f(x) = ceiling(x), is the smallest integer greater than or equal to x. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  40. Functions (2.3) (cont.) • Examples:3.5 = 3, 3.5 = 4. Note: the floor function is equivalent to truncation for positive numbers. • Example: Suppose f: B  C, g: A  B and f  g is injective. What can we say about f and g? • We know that if a  b then f(g(a))  f(g(b)) since the composition is injective. • Since f is a function, it cannot be the case that g(a) = g(b) since then f would have two different images for the same point. • Hence, g(a)  g(b) It follows that g must be an injection. However, f need not be an injection (you show). CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  41. Sequences and Summations (2.4) • Definition: A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2, . . . } to a set S. Note: the sets {0, 1, 2, 3, . . . , k} and {1, 2, 3, 4, . . . , k} are called initial segments of N. Notation: if f is a function from {0, 1, 2, . . .} to S we usually denote f(i) by ai and we write where k is the upper limit (usually ). CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  42. Sequences and Summations (2.4) (cont.) Examples: Using zero-origin indexing, if f(i) = 1/(i + 1). then the Sequence f = {1, 1/'2,1/3,1/4, . . . } = {a0, a1, a2, a3, . . } Using one-origin indexing the sequence f becomes {1/2, 1/3, . . .} = {a1, a2, a3, . . .} CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  43. Sequences and Summations (2.4) (cont.) • Summation Notation Given a sequence we can add together a subset of the sequence by using the summation and function notation or more generally CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  44. Sequences and Summations (2.4) (cont.) Examples: Similarity for the product notation:

  45. Sequences and Summations (2.4) (cont.) Definition:A geometric progression is a sequence of the form a, ar, ar2, ar3, ar4, . . . . Your book has a proof that (you can figure out what it is if r = 1). You should also be able to determine the sum • if the index starts at k vs. 0 • if the index ends at something other than n (e.g., n-1, n+1, etc.). CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  46. Sequences and Summations (2.4) (cont.) • Cardinality • Definition:The cardinality of a set A is equal to the cardinality of a set B, denoted | A | = | B |, if there exists a bijection from A to B. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  47. Sequences and Summations (2.4) • Definition:If a set has the same cardinality as a subset of the natural numbers N, then the set is called countable. If |A| = |N|, the set A is countably infinite. The (transfinite) cardinal number of the set N is aleph null = 0. If a set is not countable we say it is uncountable. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  48. Sequences and Summations (2.4) • Examples: The following sets are uncountable (we show later) • The real numbers in [0, 1] • P(N), the power set of N • Note: With infinite sets proper subsets can have the same cardinality. This cannot happen with finite sets. Countability carries with it the implication that there is a listing of the elements of the set. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  49. Sequences and Summations (2.4) • Definition: | A |  | B | if there is an injection from A to B. Note: as you would hope, • Theorem: If | A |  | B | and | B |  | A | then | A | = | B |. This implies • if there is an injection from A to B • if there is an injection from B to A then • there must be a bijection from A to B • This is difficult to prove but is an example of demonstrating existence without construction. • It is often easier to build the injections and then conclude the bijection exists. CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

  50. Sequences and Summations (2.4) • Example: Theorem: If A is a subset of B then | A |  | B |. Proof: the function f(x) = x is an injection from A to B. • Example: {0, 2, 5}| 0 The injection f: {0, 2, 5}  N defined by f(x) = x is shown below: 0 1 2 3 4 5 6 … 0 2 5 CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

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