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2.1 Sets ‒ Sets ‒ Common Universal Sets ‒ Subsets 2.2 Set Operations 2.3 Functions 2.4 Sequences and Summations 1 Sets • 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. 2 Sets • 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. 3 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+ ={1,2,3,…},the set of positive integers. • Q={P/q | p Z,q Z and q≠0},the set of rational numbers. • Q+={x R | x=p/q, for some positive integers p and q} 4 Common Universal Sets • Notation: x is a member of S or x is an element of S: x S x is not an element of S: xS 5 Subsets • Definition: The set A is a subset of the set B, denoted A B, iff x[xA→xB] • Definition: The void set, the null set, the empty set, denoted Ø, is the set with no members. • Note: the assertion x Ø is always false. Hence x[x Ø → x B] is always true. Therefore, Ø is a subset of every set. • Note: Set B is always a subset of itself. 6 Subsets • Definition: If A B but A B the we say A is a proper subset of B, denoted A B. • Definition: The set of all subset of set A, denoted P(A), is called the power set of A. 7 Subsets • Example: If A = {a, b} then P(A) = {Ø, {a}, {b}, {a,b}} 8 Subsets • 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. 9 Subsets • 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 • 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. 10 Subsets • Example: A = {Ø,{Ø}}. A has two elements and hence four subsets: Ø , {Ø} , {{Ø}} , {Ø,{Ø}} Note that Ø is both a member of A and a subset of A! 11 P. 1 Subsets • Example: A = {Ø}. Which one of the follow is incorrect: (a) ØA (b) ØA (c) {Ø} A (d) {Ø} A 12 P. 1 Subsets • 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? 13 P. 1 Subsets • Definition: The Cartesian product of A with B, denoted AB, is the set of ordered pairs {<a, b> | a A Λ b B} n Notation: Ai ={<a1,a2,...,an>|ai Ai} i 1 • Note: The Cartesian product of anything with Ø is Ø. (why?) 14 P. 1 Subsets • 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|? 15 P. 1 Terms • Sets • Subset 16 P. 1 2.1 Sets 2.2 Set Operations – – – – Set Operations Venn Diagrams Set Identities Union and Intersection of Indexed Collections 2.3 Functions 2.4 Sequences and Summations 17 P. 1 Set Operations • 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. 18 P. 1 Set Operations • 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) Λ (x B → x A)] or A = B iff A B and B A 19 P. 1 Set Operations • Definitions: – The union of A and B, denoted AB, is the set {x | xA V xB} – The intersection of A and B, denoted AB, is the set {x | xA Λ xB} FIGURE 1 ∪ FIGURE 2 ∩ 20 P. 1 Set Operations • Note: If the intersection is void, A and B are said to be disjoint. – The complement of A, denoted A , is the set {x | x A}={x | ¬(xA)} • Note: Alternative notation is A C, and {x|xA}. – The difference of A and B, or the complement of B relative to A, denoted A - B, is the set A B • Note: The complement of A is U - A. – The symmetric difference of A and B, denoted AB, is the set (A - B)(B - A). 21 P. 1 Set Operations • 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 – – – – – – – AB = {1, 2, 3, 4, 5, 6, 7, 8} AB = {4, 5} A = {0, 6, 7, 8, 9, 10} B = {0, 1, 2, 3, 9, 10} A - B = {1, 2, 3} B - A = {6, 7, 8} A B = {1, 2, 3, 6, 7, 8} 22 P. 1 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. 23 P. 1 Set Identities • Set identities correspond to the logical equivalences. • Example: The complement of the union is the intersection of the complements A B A B 24 P. 1 Set Identities • Proof: To show x[ x A B x A B] 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. • We now apply an important rule of inference (defined later) called Universal Instantiation 25 Set Identities • 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. 26 Set Identities • We say 'Let x be arbitrary. ' Then we can treat the predicates as propositions: Assertion Reason Def. of complement x A B x [A B] Def. of x A B ¬[ x A B] ¬[ x A x B] Def. of union ¬ xA Λ ¬ x B DeMorgan's Laws Def. of x A xB Def. of complement x A xB Def. of intersection x A B Hence x A B x A B is a tautology. 27 P. 1 Set Identities Since – x was arbitrary – we have used only logically equivalent assertions and definitions 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 A B x A B ] P. 1 Set Identities • Q. E. D. (an abbreviation for the Latin phrase “Quod Erat Demonstrandum” - “which was to be demonstrated” used to signal the end of a proof) • Note: As an alternative which might be easier in some cases, use the identity A B [A B and B A] 29 Set Identities • Example: Show A(B - A) = Ø The void set is a subset of every set. Hence, A (B - A) Ø Therefore, it suffices to show A(B - A) Ø or x[x A (B - A) → x Ø] 30 P. 1 Set Identities So as before we say 'let x be arbitrary’. Show x A (B- A) → x Ø is a tautology. But the consequent is always false. Therefore, the antecedent better always be false also. 31 Set Identities • Solution: Apply the definitions: Assertion x A(B- A)x A Λ x (B- A) x A Λ (x B Λ x A) (x A Λ x A) Λ x B 0ΛxB 0 (false) Reason by Def. of by Def. of by Def. of Associative by Def. of Complement by Def. of Domination 32 P. 1 Union and Intersection of Indexed Collections • Let A1,A2 ,..., An be an indexed collection of sets. • Definition: The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection. we use the notation n i 1 Ai A1 A2 ... An to denote the union of the sets A1,A2 ,..., An 33 P. 1 Union and Intersection of Indexed Collections • Definition: The intersection of a collection of sets is the set that contains those elements that are members of all the set in the collection. we use the notation n i 1 Ai A1 A2 ... An to denote the intersection of the sets A1,A2 ,..., An 34 Union and Intersection of Indexed Collections • Examples: • Let Ai [i, ),1 i n Ai [1, ) n Ai [n, ) i 1 i 1 35 P. 1 Terms • • • • • • Boolean Algebra Set operations Union Intersection Complement Difference • • • • • Symmetric difference Venn Diagram Set Identities Universal Instantiation Universal Generalization 36 P. 1 2.1 Sets 2.2 Set Operations 2.3 Functions ‒ Functions ‒ Injections, Surjections and Bijections ‒ Inverse Functions ‒ Composition 2.4 Sequences and Summations 37 P. 1 Functions • 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 38 P. 1 Functions • Note: f associates with each x in A one and only one y in B. – A is called the domain – 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). 39 P. 1 Functions • The range of f is the set of all images of points in A under f. We denote it by f(A). • If S is a subset of A then f(S) = {f(s) | s in S}. • Q: What is the difference between range and codomain? 40 P. 1 Functions • Example: 1. 2. 3. 4. 5. 6. 7. f(a) = the image of d is the domain of f is the codomain is f(A) = the preimage of Y is the preimages of Z are 8. f({c,d}) = Z Z A = {a, b, c, d} B = {X, Y, Z} {Y, Z} b a, c ,d {Z} 41 P. 1 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). 42 P. 1 Injections, Surjections and Bijections • Examples: The previous Example function is neither an injection nor a surjection. Hence it is not a bijection. 43 P. 1 Injections, Surjections and Bijections • 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. 44 P. 1 Injections, Surjections and Bijections • Examples: Let A = B = R, the reals. Determine which are injections, surjections, bijections: 1. 2. 3. 4. 5. f(x) = x, f(x) = x2, f(x) = x3, f(x) = | x |, f(x) = x + sin(x) 45 P. 1 Injections, Surjections and Bijections • 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. (見山是山,見水是水前,會先見山不是山,見 水不是水) 46 P. 1 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 47 P. 1 Inverse Functions • Example: Let f be defined by the diagram: • Note: No inverse exists unless f is a bijection. 48 P. 1 Inverse Functions • 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. 49 P. 1 Inverse Functions • Example: Let f be the following function: f-1({Z}) = f-1({X, Y}) = {c, d} {a, b} 50 P. 1 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)) 51 P. 1 Composition • Examples: • If f(x) =x 2 and g(x) = 2x + 1, then f(g(x)) =(2 x 1) 2 and g(f(x)) = 2x 2 + 1 52 P. 1 Floor and Ceiling • 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. • Examples: 3.5 = 3, 3.5 = 4. • Note: the floor function is equivalent to truncation for positive numbers. 53 P. 1 Composition • Example: Suppose f: B→C, g: A→B and fg 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). 54 P. 1 Terms • • • • • Function Image Preimage Range Injective (one-toone) • Surjective • Bijective • • • • • Cardinality Inverse function Composition Floor function Ceiling function 55 P. 1 2.1 2.2 2.3 2.4 ‒ ‒ ‒ ‒ ‒ Sets Set Operations Functions Sequences and Summations Sequences and Summations Summation Notation Cardinality Some Countably Infinite Sets Cantor Diagonalization P. 1 Sequences and Summations • 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. P. 1 Sequences and Summations • Notation: if f is a function from {0, 1, 2, . . .} to S we usually denote f(i) by ai and we k k { a , a , a , a ,...} { a } { a } write 0 1 2 3 i i 0 i 0 where k is the upper limit (usually ∞). P. 1 Sequences and Summations • 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, . . .} P. 1 Summation Notation k { a } • Given a sequence i 0 we can add together a subset of the sequence by using the summation and function notation n ag ( m) ag ( m1) ... ag ( n ) ag ( j ) j m or more generally a jS j P. 1 Summation Notation • Examples: n r r r r ... r r j 0 1 2 3 n 0 • If S = {2, 5, 7, 10} then a jS 1 1 1 1 1 ... 2 3 4 1 i n a2m a2( m1) ... a2( n) a2 j j m j a2 a5 a7 a10 • Similarly for the product notation: n a j am am 1...an j m P. 1 Summation Notation • Definition: A geometric progression is a sequence 2 3 4 a , ar , ar , ar , ar ,.... of the form • Your book has a proof that n 1 n r 1 i r if r 1 r 1 i 0 (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.). P. 1 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. • 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 = ﭏ0. If a set is not countable we say it is uncountable. P. 1 Cardinality • 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. P. 1 Cardinality • 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. P. 1 Cardinality • 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} | ≤ ﭏ0 The injection f: {0, 2, 5} →N defined by f(x) = x is shown below: Some Countably Infinite Sets • The set of even positive integers E ( 0 is considered even) is countably infinite. Note that E is a proper subset of N! • Proof: Let f(x) = 2x. Then f is a bijection from N to E • Z+, the set of positive integers is countably infinite. P. 1 Some Countably Infinite Sets • The set of positive rational numbers Q+ is countably infinite. The position on the path (listing) indicates the image of the bijective function f from N to QR: • f(0) = 1/1, f(1) = 1/2, f(2) = 2/1, f(3) = 3/1, and so forth. • Every rational number appears on the list at least once, some many times (repetitions). • Hence, | N | = | QR | = ﭏ0. • Q. E. D. • The set of all rational numbers Q, positive and negative, is countably infinite. P. 1 Some Countably Infinite Sets • The set of (finite length) strings S over a finite alphabet A is countably infinite. • To show this we assume that - A is nonvoid - There is an “alphabetical” ordering of the symbols in A • Proof: List the strings in lexicographic order: – all the strings of zero length, – then all the strings of length 1 in alphabetical order, – then all the strings of length 2 in alphabetical order, etc. • This implies a bijection from N to the list of strings and hence it is a countably infinite set. P. 1 Some Countably Infinite Sets • For example: Let A = {a, b, c}. Then the lexicographic ordering of A is { , a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, aab, aac, aba, ....} ={f(0), f(1), f(2), f(3), f(4), . . . .} P. 1 Some Countably Infinite Sets • The set of all C programs is countable . • Proof: Let S be the set of legitimate characters which can appear in a C program. – A C compiler will determine if an input program is a syntactically correct C program (the program doesn't have to do anything useful). – Use the lexicographic ordering of S and feed the strings into the compiler. – If the compiler says YES, this is a syntactically correct C program, we add the program to the list. – Else we move on to the next string. In this way we construct a list or an implied bijection from N to the set of C programs. Hence, the set of C programs is countable. Q. E. D. P. 1 Cantor Diagonalization • An important technique used to construct an object which is not a member of a countable set of objects with (possibly) infinite descriptions. • Theorem: The set of real numbers between 0 and 1 is uncountable. • Proof: We assume that it is countable and derive a contradiction. If it is countable we can list them (i.e., there is a bijection from a subset of N to the set). We show that no matter what list you produce we can construct a real number between 0 and 1 which is not in the list. Hence, there cannot exist a list and therefore the set is not countable P. 1 Cantor Diagonalization It's actually much bigger than countable. It is said to have the cardinality of the continuum, c. • Represent each real number • THE LIST.... in the list using its decimal r1 = .d11d12d13d14d15d16. . . . . expansion. e.g., 1/3 = .3333333........ r2 = .d21d22d23d24d25d26 . . . . 1/2 = .5000000........ r3 = .d31d32d33d34d35d36 . . . . = .4999999........ If there is more than one expansion for a number, it doesn't matter as long as our construction takes this into account. P. 1 Cantor Diagonalization • Now construct the number x = .x1x2x3x4x5x6x7. . . . xi = 3 if dii 3 xi = 4 if dii = 3 Then x is not equal to any number in the list. Hence, no such list can exist and hence the interval (0,1) is uncountable. Q. E. D. P. 1 Terms • • • • • • • • • Sequence Summation Cardinality Zero-origin indexing One-origin indexing Geometric progression Countable infinite Uncountable Cantor diagonalization • • • • • • • • • Sequence Summation Cardinality Zero-origin indexing One-origin indexing Geometric progression Countable infinite Uncountable Cantor diagonalization P. 1