Chapter 2 - Basics Structures
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1 Chapter 2 - Basics Structures Sets Definitions and Notation Definition 1 (Set). A set is an of. These are called the or of the set. We ll typically use uppercase letters to denote sets: S, A, B,... When listing out the elements in a set (what the book calls the roster method ) we ll use braces, e.g., S = {2, red, water, {1}} Membership: x S means x / S means Note: and do not matter when listing the elements of a set. For example, {5, 3, 2, 7, 3, 8, 2, 2, 2} is the same set as {2, 3, 5, 7, 8}. Examples Let S be the set of people in this room. Some standard sets of numbers: N is the set of, {0, 1, 2, 3,... }. Note: Some books define N to be {1, 2, 3,... } (the counting numbers ). Z is the set of all. Q is the set of all. R is the set of all. C is the set of all. Z + is the set of all (sometimes written Z + ). You may also see this sort of notation to denote other subset of certain sets of numbers, such as Z <0, Z 0, R + (or R + ), Q +, etc. Note: positive does not include 0, but saying nonnegative does include 0. Set-Builder Notation Often we will write sets using set-builder notation; the general form is {x P (x)} or {x : P (x)} for some predicate P (x). The or : is read such that. 1
2 Example {x x is a left-handed guitar player} 2. {x x R and 1 x 2} We can also use set-builder notation to make the domain(s) explicit: Example 2. {x R : x 2 < 10} {x N : x 2 < 10} {x Z : x 2 < 10} Examples 1. The set of all integers that are perfect squares. 2. {2, 4, 6} Venn Diagrams The is the set of all objects under consideration (similar to domain in the previous chapter). We denote it by U and draw a rectangle in the Venn Diagram. In a Venn Diagram we picture a set, say A, as a restricted portion of the universal set: The Empty Set Definition 2. Empty Set The empty set is the set that contains. Example 3. Is = { }? 2
3 Subsets Notation: Predicate Definition: We can define subset using predicates. Example 4. For all sets S, 1. Is S? 2. Is S S? Subsets and Equality To show that A B, we show that if then. To show that A B, we find at least one such that. Two set A and B are equal if and only if (TFAE): They both contain Sundry Definitions Proper subsets: 3
4 Finite sets: Infinite sets: Cardinality of a set: A is denotes the of A. If A is then we can just count them all and A =(some number). If A is, then we can still have. New Sets From Old The Power Set: P(S) denotes the set of. It is a. Cartesian Products: The of A and B is a set of /. We write this as A B = In general, A 1 A 2 A n = Example 5. Let B = {1, 2} and C = {a, b, c}. Find the following: 1. P( ) 5. B C 2. P ({ }) 6. B B 3. P(B) 7. P(C) 4. B C 8. P (P({ })). Note: P(A) = 4
5 2.2 - Set Operations Union of Sets Definition 3 (Union). The union of two sets A and B is the set of all elements which are Notation Venn Diagram Intersection of Sets Definition 4 (Intersection). The intersection of two sets A and B is the set of all elements which are in both A and B. Notation Venn Diagram Generalized Unions and Intersections Both unions and intersections are associative, so their generalizations are well-defined. Notation: 5
6 More Definitions Disjoint Sets: Mutually Disjoint Collections of Sets: Principle of Inclusion/Exclusion: Set Difference Set Complement Definition 5 (Complement of a Set). Venn Diagram 6
7 Set Identities Identity A U = A A = A A U = U A = A A = A A A = A (A) = A A B = B A A B = B A A (B C) = (A B) C A (B C) = (A B) C A (B C) = (A B) (A C) A (B C) = (A B) (A C) A B = Ā B A B = Ā B A (A B) = A A (A B) = A A Ā = U A Ā = Name Identity Laws Domination Laws Idempotent Laws Complementation Law Commutative Laws Associative Laws Distributive Laws De Morgan s Laws Absorption Laws Complement Laws Proving Set Identities Example 6. Prove the Second Absorption Law: A (A B) Proof. Q.E.D. 7
8 Example 7. Prove the first part of De Morgan s Law: Proof. A B = (Ā B) Q.E.D. 8
9 2.3 - Functions Definition 6 (Function). Given nonempty sets A and B, a function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b where b is the unique element of the set B to which f maps the element a of the set A. Functions are also called mappings, transformations, or assignments. Example Each student in this room is assigned to exactly one seat. 2. Each person is assigned to exactly one birth mother. 3. Each nonnegative real number is assigned exactly one square root. Notation If f is a function from A to B, we write this as f : A B. If A and B are relatively small sets, we can draw the function: More terminology Given a function f : A B, Domain and Codomain: Range: Image: Pre-Image: Arithmetic on Functions: If f 1 and f 2 are functions whose codomain is the real numbers, then we can define f 1 + f 2 and f 1 f 2 as... 9
10 Image of a Set Suppose f : A B and S A. Then f(s) = Example 9. Let A = {1, 2, 3, 4, 5} and B = {a, b, c, d}, and S = {2, 3, 4} and T = {1, 5}, and let f(1) = b, f(2) = c, f(3) = a, f(4) = d, and f(5) = b. Find 1. f( ) 2. f(s) 3. f(t ) One-to-One and Onto Definition 7 (One-to-One). A function f : A B is said to be one-to-one (abbreviated 1-1 ) or injective if and only if... Definition 8 (Onto). A function f : A B is said to be onto or surjective if and only if... 10
11 Bijection and Inverse Definition 9 (Bijection). A function f : A B is said to be a bijection or one-to-one correspondence if and only if it is both injective and surjective. Definition 10 (Inverse). The inverse (when it exists) of a function f : A B is the function... Cardinality and Functions If f : A B is 1-1, then A B. If f : A B is onto, then A B. If f : A B is a bijection, then A B. Monotonic Functions A function f is said to be increasing if and only if x < y implies that... A function f is said to be decreasing if and only if x < y implies that... A function f is said to be strictly increasing if and only if x < y implies that... A function f is said to be strictly decreasing if and only if x < y implies that... 11
12 Proofs Involving Functions Example 10. Show that f : R R defined by f(x) = 3x + 4 is a bijection. Proof. Q.E.D. Example 11. Let A be the set of even integers and B be the set of odd integers. Define f : A B as f(x) = x + 1. Determine whether f is a bijection. Proof. Q.E.D. 12
13 Compositions of Functions Definition 11. Given two functions f : B C and g : A B, we define the composition of f and g to be... The Graph of a Function Definition 12. The graph of a function f : A B is the set of all ordered pairs (a, b) for which f(a) = b. One last thing... Definition 13 (Floor and Ceiling Functions). x x Example
14 2.4 - Sequences and Summations Sequences Definition 14 (Sequence). A sequence is a function from some subset of the integers (usually N or Z + ) into R. Instead of writing f(i) for the function value, we instead use subscripts to denote the ith function value and write a i. Notation: The notation for an infinite sequence looks like a 0, a 1, a 2,... or = {a i } or = {a i } i=0 Note: Although we use braces as our delimiters, a sequence is not the same as a set because the order of a sequence matters! Unless otherwise noted: if the function is defined at 0 then we assume our sequence starts at the index 0; if the function is not defined at 0 then we assume the sequence starts at the index 1. Progessions Definition 15 (Geometric Progression). A geometric progression is a sequence of the form a, ar, ar 2,..., ar n,... where a, r R. a is called the initial term and r is called the common ratio. Definition 16 (Arithmetic Progression). An arithmetic progression is a sequence of the form a, a + d, a + 2d,..., a + nd,... where a, d R. a is called the initial term and d is called the common difference. Example 13. Find the pattern in each of the following: 1. 3, 10, 31, 94, , 1 4, 1 8, 1 16, , 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4,... Definition 17 (Strings). Finite sequences of the form a 1, a 2,..., a n are sometimes viewed and written as strings: The empty string is denoted by λ. a 1 a 2... a n 14
15 Summations and Summation Notation The sum of the first n terms of a sequence {a n } is denoted by... Reindexing: Suppose we want to change the starting index of a sum... Theorem 1 (Finite Geometric Series). The sum of a finite geometric series (finite geometric progression) is n a ar n+1 ar i if r 1 = 1 r i=0 (n + 1)a if r = 1 Other Useful Sums n k = k=1 k=0 n(n + 1) 2 ar k = a 1 r for r < 1 (See the book for other summation formulae.) Summation Over Members of a Set Example 14. Let A = {0, 2, 4}, f : A R given by f(i) = a i = 3 i. Then a i = i A 15
16 Double Summation 3 i=1 j=1 4 (i j) Recurrence Relations Definition 18 (Recurrence Relation). A recurrence relation for the sequence {a n } is a recursive definition for the terms of the sequence which expresses a n in terms of one or more of the previous terms of the sequence. Definition 19 (Solution). A solution of a recurrence relation is a sequence whose terms satisfy the recurrence relation. A closed formula solution of a recurrence relation is a non-recursive solution of the recurrence relation. Definition 20 (Initial Conditions). The initial conditions of a recursively defined sequence specify the terms that precede the first terms where the recurrence relation takes effect. Example 15. Let a 0 = 1, a n = 3a n Find the first 5 terms in the sequence, and show that a n = 3n is a closed formula solution. 16
17 Example 16. Determine whether the sequences a n = 2 n and a n = n 4 n are solutions of the recurrence relation a n = 8a n 1 16a n 2 The Towers of Hanoi Suppose we have n disks of different sizes and three pegs, and the disks are stacked on one of the pegs 1 in order of size with the largest at the bottom of the peg. If we are only allowed to move one disk at a time and cannot put a larger disk on a smaller one, then how many moves are needed to move the entire tower to a new peg? 1 Drill a hole in each disk, if needed. 17
18 2.5 - Cardinality Definition 21. Two sets A and B have the same cardinality if and only if there exists a bijection between them. When this is the case, we write A = B. (Note: This definition holds for any sets A and B not just finite sets.) Definition 22. If there is a 1-1 function from A to B, then the cardinality of A is less than or equal to the cardinality of B, and we write A B. When A B and A and B have different cardinality, then the cardinality of A is less than the cardinality of B, and we write A < B. Definition 23. A set is countable if it is finite or there exists a bijection between the set and Z +. A set that is not countable is uncountable. When an infinite set S is countable, we denote the cardinality of S by ℵ 0 (read aleph-null or aleph-nought ) and write S = ℵ 0. Example A graphical proof that Q has cardinality ℵ 0 : 18
19 2.6 - Matrices Definitions Definition 24 (Matrix, pl. Matrices). A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m n (read m by n ) matrix; if m = n then we say that the matrix is a square matrix. Two matrices are equal if and only if the corresponding entries in every position are equal. Example 17. Notation: The book uses boldface uppercase letters to denote matrices; normal font uppercase letters are typically used in handwritten work and also in some textbooks. The book uses brackets as delimiters, but parentheses are also commonly used. Definition 25. Let a 11 a a 1n a 21 a a 2n A =... a m1 a m2... a mn The ith row of A is the 1 n row matrix [ ] a 2j a i1 a i2... a in ; the jth column of A is the m 1 column matrix. a mj The ijth element of A is denoted by a ij, and we may write A = [a ij ]. a 1j Matrix Addition Definition 26. Let A = [a ij ] and B = [b ij ] both be m n matrices. The sum of A and B is the matrix A+B = [a ij +b ij ]. So, we simply add corresponding elements of each matrix. Note: If the matrices are different sizes, then the sum is undefined! Example 18. Let A = [ ] 1 2 3, B = [ ] 1 0 1, C =
20 Matrix Scalar Multiplication Definition 27. Let A = [a ij ] and α R. Then i.e., we multiply each element of the matrix by α. αa = [α a ij ], 1 0 Example 19. Let α = 2, C = Matrix Multiplication Definition 28. Let A = [a ij ] be an m k matrix and B = [b ij ] be a k n matrix. The product of A and B is AB = [c ij ], where c ij = a i1 b 1j + a i2 b 2j + + a ik b kj 2 Note: Matrix multiplication is not commutative! Example 20. Let A = [ ] 1 2 3, B = [ ] 1 0 1, C = Definition 29 (Identity). The identity matrix is the n n matrix I n which has 1s on the diagonal and 0s everywhere else: I n = If we multiply a matrix A by an appropriately sized identity matrix, then the product is still A. 2 This is the dot product of the ith row of A with the jth column of B. 20
21 Definition 30 (Inverse). The inverse of a square n n matrix is the n n matrix A 1 such that AA 1 = A 1 A = I n For 2 2 matrices we have the following formula for the inverse: [ ] 1 [ ] a b 1 d b = c d ad bc c a Example 21. Transpose of a Matrix Definition 31 (Transpose). Let A = [a ij ] be an m n matrix. The transpose of A is the n m matrix A t = [a ji ] This means that the rows of A become the columns of A t (this also means that the columns of A become the rows of A t ). Example 22. Definition 32 (Symmetric). A matrix A is said to be symmetric if Note: If A is symmetric then it must be square. A = A t Example 23. Zero-One Matrices A zero-one matrix is one whose entries are all either zeros or ones. We can combine zero-one matrices using Boolean operations: Meet: A B Join: A B Boolean product: A B 21
22 Powers of a Matrix If A is a square matrix, then we can define I n if r = 0 A r = AA }{{... A } if r Z + r times If A is a square zero-one matrix, then we can define the rth Boolean power of A as I n if r = 0 A [r] = A A A }{{} if r Z + r times Example 24. A =
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