Basic Structures: Sets, Functions, Sequences, and Sums

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Sec 2.1 . Sets. Definitions: Set; element of; contains; ?. The objects in the set are called elements or members of the set.A set is said to contain its elements.The empty set ? is a set which contains no elements.The universal set, U is the set of all elementss under consideration. Standar

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Basic Structures: Sets, Functions, Sequences, and Sums

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

2. Sec 2.1 Sets

3. Definitions: Set; element of; contains; Ø The objects in the set are called elements or members of the set. A set is said to contain its elements. The empty set Ø is a set which contains no elements. The universal set, U is the set of all elementss under consideration

4. Standard Sets N - Set of natural numbers,{0,1,2,3,…} Z - Set of integers,{…,-2,-1,0,1,2,…} Z+ - Set of positive integers,{1,2,3,…} Q - Set of rational numbers,{p/q|p?Z,q?Z,q?0} R - Set of real numbers

5. Definitions: Set equality; Subset; Finite and Infinite Cardinality Two sets are equal if they have the same elements. Set A is said to be a subset of set B (A ? B) if every element of A is also an element of set B. Set A is a proper subset of set B (A ? B) if A ?B and A ? B. A set with n distinct elements is said to be a finite set. |S| - The cardinality of set S is the number n of elements is the set. A set that is not finite is called infinite

6. Definitions: Power Set, Ordered n-tuple; Cartesian Product P(S) - The power set of set S is the set of all subsets of S. The ordered n-tuple (a1, a2, … , an) is the ordered collection where for each i, ai ? Ai. A x B – The Cartesian product of sets A and B is the set of all ordered pairs (a,b) with a ? A and b ? B. A1 x A2 X … x An - The Cartesian product of sets A1, A2, … , An is the set of all ordered n-tuples (a1, a2, … , an) where ai ? Ai. ?

7. Homework Sec 2.1 pg. 119 # 1,3,5,7,13,17,21, 23, 28, 29

8. Sec 2.2 Set Operations

9. Definitions: Union; Intersection; Empty set; Disjoint Sets A?B – The union of sets A and B is the set of all elements that are contained in either A or B or both. A?B - The intersection of sets A and B is the set of all elements that are contained in both A and B. ? - The empty set is the set with no elements. Disjoint – Two sets are disjoint if the intersection of these two sets is the empty set. ?

10. Definitions: Universal Set; Complement, Difference Universal set, U: The universal set contains all elements under consideration. A :The complement of set A is the set of all elements in the Universal set that are not in A. B – A: The difference of B and A (or the complement of A relative to B) is the set of all elements in B, except those in A, or equivalently B ? A. ?

11. Set Properties Identity Law: A ? ? = A, A ? U = A Domination Law: A ? U = U, A ? ? = ? Idempotent Laws: A ? A = A, A ? A = A Complementation Laws: (Ac)c = A Note here: Ac =A Commutative Laws: A ? B = B ? A, A ? B = B ? A Associative Laws: A ? (B ? C) = (A ? B) ? C, A ? (B ? C) = (A ? B) ? C Distributive Laws: A ? (B ? C) = (A ? B) ? (A ? C) A ? (B ? C) = (A ? B) ? (A ? C) De Morgan’s Laws: (A ? E)c = A ? E, (A ? E)c = A ? E Absorption Laws: A ? (A ? B) = A, A ? (A ? B) = A Complement Laws: A ? A = U, A ? A = ?

12. Proving Set Identities To prove that two sets A and B are equal: Method I: Show that A ? B and B ? A. That is take an element x from A and, using logic, verify that x is in B and conversely argue that if x?B then x?A. Method II: If set A and B are formed by combining sets, use a set membership table to show that sets A and B have identical columns in the table.

13. Generalized Unions & Intersections Generalized union of a collection of sets is the sets that contains those elements that are members of at least one set in the collection. Notation: Generalized intersection of a collection of sets is the sets that contains those elements that are members of all the sets in the collection. Notation:

14. Computer Representation of Sets Universal set U with the bit string of length n a1,a2,…,an Subset A of U is represented a bit string with 1 if ai belongs to A; 0 if ai not eg: U={1,2,3,4,5,6}, A={1,3,5}, A is 101010 Boolean Operations: 1?1=1; 1?0=0; 1?1=1; 1?0=1 eg: 101 ? 011=001; 101 ? 011= 111

15. Homework Sec 2.2 pg. 130 # 1,3,11,13,15,17,23,29,49(a,b), 50, 51

16. Section 2.3 Functions

17. Definition: function f: A ? B: A function, f, is a correspondence between two sets, A and B, such that for each element of set A there corresponds exactly one element of the B. Notation: f(a) = b denotes the fact that the function makes the assignment between the value a ? A and the value b ?B If f: A ? B then we say f maps A to B. ?

18. Definitions: Domain; Codomain; Image; Pre-image; Range: Maps The domain of the function f:A?B is the set A. The codomain is the set B. If f(a) = b, then b is called the image of a and a is called the pre-image of b. If S is a subset of A, the image of S is the subset of B that contains all the images of elements of S. The Range of f is the set of all images of elements of A. The function f:A?B is said to map set A to set B. ?

19. Increasing/Decreasing Functions Definition: Let f be a function whose domain and codomain are subsets of the real numbers and suppose x and y are in the domain of f. f is said to be increasing if f(x) ? f(y) whenever x < y. f is said to be strictly increasing if f(x) < f(y) whenever x < y. f is said to be decreasing if f(x) ? f(y) whenever x < y. f is said to be strictly decreasing if f(x) > f(y) whenever x < y.

20. Definitions: Injection, Surjection, Bijection Let f: A ? B be a function. f is injective or one-to-one iff f(x) = f(y) implies that x = y for all x and y in A. f is surjective or onto if set B is the image of A (i.e. ?b ?B ? a?A such f(a) = b.) f is a bijection or a one-to-one correspondence if f is both surjective and injective (i.e. it is both one-to-one and onto) ?

21. Definitions: Inverse Let f:A?B be a one-to-one correspondence from set A to set B. The inverse of f is the function that assigns to each b?B the unique element a?A such that f(a) = b. The inverse function is denoted by f-1. Note: f-1(b) = a if and only if f(a) = b, thus: ?y?B f(f-1(b)) = b and ?x?A f-1(f(a)) = a. ?

22. Definition: Composition of two functions, Graphs Let g: A ? B and f: B ? C be two functions. The composition of f and g denoted by f o g is defined by f o g (x) = f(g(x)). The domain of the function is the set of x in the domain of f such that g(x) is in the domain of f. The graph of a function f: A?B is the set of all ordered pairs {(a,b)| a?A and f(a)=b} ?

23. Graph, ceiling, floor Let f:A ? B. The graph of the function f is the set of ordered pairs {(a,b)|a?A & b=f(a)} Ceiling : f(x) =+x+ The ceiling function assigns to the real number x the smallest integer that is greater than or equal to x. Floor : f(x) = +x+ The floor function assigns to the real number x the largest integer that is less than or equal to x. Factorial Function: f:N ?Z+, f(n)=n!=1*2*3…*n Define: f(0)=0!=1

24. Useful results for Floor & Ceiling Function 1a. +x+=n iff n ?x<n+1 1b. +x+=n iff x-1<n ?x 1c. +x+=n iff n-1<x ?n 1d. +x+=n iff x ?n<x+1 2. x-1< +x+ ? x? +x+ <x+1 3a. +-x+ = -+x+ 3b. +-x+ = - +x+ 4a. +x+n+ = +x++n 4b. +x+n+ = +x+ +n

25. Homework Sec 2.3 pg. 133 # 1,9,10,11,19,23,27,32,33

26. Sec 2.4 Sequences and Summation

27. Definitions Sequence {an}: A sequence is a function whose domain is either the set {0, 1, 2, …} or the set {1, 2, 3,…} and whose codomain is a generally a set of numbers. We use the notation an to denote the image of the integer n and we call an a term of the sequence. Geometric Progression: A geometric progression is a sequence of the form: a, ar, ar2, …, arn. The number a is called the initial term and the number r is called the common ratio. Arithmetic Progression: An arithmetic progression is a sequence of the form: a, a+d, a+2d, …, a+nd. The number a is called the initial term and the number d is called the common difference. ?

28. Summation Notation

29. Geometric Progression Theorem

30. Useful Summation Results

31. Definitions Cardinality: The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B. A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. ?

32. Theorems Theorem: The set of all Rational numbers is countable Theorem: The set of all real numbers in the interval [0,1] are not countable.

33. Homework Sec 2.4 pg. 161 # 3, 5, 7, 13, 15, 17, 31

34. THE END Chapter 2

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