Why Study Discrete Math? Digital computers are based on discrete - PowerPoint PPT Presentation

Why Study Discrete Math? ■ Digital computers are based on discrete units Sets of data (bits). (Rosen, Sections 2.1,2.2) ■ Therefore, both a computer’s ■ structure (circuits) and TOPICS ■ operations (execution of algorithms) • Discrete math can be described by discrete math • Set Definition • Set Operations ■ A generally useful tool for rational • Tuples thought! Prove your arguments. CS 160, Summer Semester 2016 2 Uses for Discrete Math in What is ‘discrete’? Computer Science ■ Consisting of distinct or unconnected elements, not Advanced algorithms & data structures ■ continuous (calculus) Programming language compilers & interpreters. ■ ■ Helps us in Computer Science: Computer networks ■ ■ What is the probability of winning the lottery? Operating systems ■ ■ How many valid Internet address are there? Computer architecture ■ How can we identify spam e-mail messages? ■ ■ How many ways are there to choose a valid password on our Database management systems ■ computer system? Cryptography ■ How many steps are needed to sort a list using a given ■ method? Error correction codes ■ ■ How can we prove our algorithm is more efficient than Graphics & animation algorithms, game engines, etc. … another? ■ i.e. , the whole field! ■ CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 3 4

What is a set? What is a set? ■ An unordered collection of objects ■ Objects are called elements or members of the set ■ {1, 2, 3} = {3, 2, 1} since sets are unordered. ■ Notation ∈ ■ {a, b, c} = {b, c, a} = {c, b, a} = {c, a, b} = {a, c, b} ■ {2} ■ a ∈ B means “a is an element of set B.” ■ {on, off} ■ Lower case letters for elements in the set ■ { } ■ Upper case letters for sets ■ If A = {1, 2, 3, 4, 5 } and x ∈ A, what are the possible values of x? CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 5 6 What is a set? What is a set? Infinite Sets (without end, unending) ■ Infinite vs. finite ■ N = {0, 1, 2, 3, …} is the Set of natural numbers ■ ■ If finite, then the number of elements is called Z = {…, -2, -1, 0, 1, 2, …} is the Set of integers ■ the cardinality , denoted | S | Z+ = {1, 2, 3, …} is the Set of positive integers ■ Finite Sets (limited number of elements) ■ V = {a, e, i, o, u} |V| = 5 ■ V = {a, e, i, o, u} is the Set of vowels ■ ■ F = {1, 2, 3} |F| = 3 O = {1, 3, 5, 7, 9} is the Set of odd #’s < 10 ■ ■ B = {0,1} |B| = 2 F = {a, 2, Fred, New Jersey} ■ Boolean data type used frequently in programming ■ S = {spring, summer, fall, winter} |S| = 4 ■ B = {0,1} ■ B = {false, true} ■ Seasons = {spring, summer, fall, winter} ■ ClassLevel = {Freshman, Sophomore, Junior, Senior} ■ CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 7 8

Example sets Venn Diagram ■ Alphabet ■ Graphical representation of ■ All characters set relations: ■ Booleans: true, false ■ Numbers: ■ N = {0,1,2,3…}- Natural numbers A B ■ Z = {…,-2,-1,0,1,2,…} - Integers ■ Q = - Rationals ■ R , Real Numbers ■ Note that: ■ Q and R are not the same. Q is a subset of R . ■ N is a subset of Z . U CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 9 10 What is a set? Equality ■ A = B is used to show set equality ■ Defining a set: ■ Option 1: List the members ■ Two sets are equal when they have ■ Option 2; Use a set builder that defines set of x that exactly the same elements hold a certain characteristic ■ Thus for all elements x, x belongs to A ■ Notation: {x ∈ S | characteristic of x} if and only if (iff) x also belongs to B ■ Examples: ■ A = { x ∈ Z + | x is prime } – set of all prime ■ The if and only is a bidirectional positive integers implication that we will study later ■ O = { x ∈ N | x is odd and x < 10000 } – set of odd natural numbers less than 10000 CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 11 12

Set Operations: Union Set Operations: Intersection • Operations that take as input sets and • The intersection of sets A and B is the set have as output sets containing those elements in both A and B. • The union of the sets A and B is the set that contains those elements that are either in A or • Notation: A∩B in B, or in both. • The sets are disjoint if their intersection – Notation: A ∪ B produces the empty set. – Example: union of {1, 2, 3} and {1, 3, 5} is? • Example: {1, 2, 3} intersection {1, 3, 5} is? Answer: {1, 2, 3, 5} Answer: {1, 3} CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 13 14 Set Operations: Complement Set Operations: Difference • The difference of A and B is the set of • The complement of set A is the elements that are in A but not in B. complement of A with respect to U, the universal set. • Notation: A - B • Notation: • Aka the complement of B with respect to A • Example: If N is the universal set, what is • Can you define difference using union, the complement of {1, 3, 5}? complement and intersection? Answer: {0, 2, 4, 6, 7, 8, …} • Example: {1, 2, 3} difference {1, 3, 5} is? Answer: {2} CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 15 16

Identities Subsets ■ The set A is a subset of B iff for all elements x of A, x is also an element of B. But not necessarily the reverse… ■ Notation: A ⊆ B ■ {1,2,3} ⊆ {1,2,3} ■ {1,2,3} ⊆ {1,2,3,4,5} ■ What is the relationship of the cardinality |A| <= |B| between sets if A ⊆ B ? CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 17 18 Subset Empty Set ■ Subset is when a set is contained in ■ Empty set has no elements and therefore is another set. Notation: ⊆ the subset of all sets: { } or Ø ■ Proper subset is when A is a subset of B , ■ Is Ø ⊆ {1,2,3}? - Yes! but B is not a subset of A . Notation: ⊂ ■ The cardinality of Ø is zero: | Ø | = 0. ■ ∀ x ((x ∈ A) → (x ∈ B)) ∧ ∃ x ((x ∈ B) ∧ (x ∉ A)) ■ All values x in set A also exist in set B ■ Consider the set containing the empty set: ■ … but there is at least 1 value x in B that is not in A { Ø } ■ A = {1,2,3}, B = {1,2,3,4,5} A ⊂ B, means that |A| < |B|. ■ Yes, this is indeed a set: except for infinite sets, e.g., N ⊂ Z, but |N| = |Z| = infinity Ø ∈ { Ø } and Ø ⊆ { Ø }. CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 19 20

Set Theory Powerset Quiz time: ■ The powerset of a set is the set containing all the subsets of that set. 2001 • A = { x ∈ N | x ≤ 2000 } What is |A|? Infinite ■ Notation: P (A) is the powerset of set A. • B = { x ∈ N | x ≥ 2000 } What is |B|? Yes ■ Fact: | P ( A ) | = 2 | A | . • Is {x} ⊆ {x}? ■ If A = { x, y }, then P (A) = { ∅ , {x}, {y}, {x,y} } Yes • Is {x} ∈ {x,{x}}? ■ If S = {a, b, c}, what is P ( S )? Yes • Is {x} ⊆ {x,{x}}? No • Is {x} ∈ {x}? CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 21 22 Powerset example Example • Number of elements in powerset = 2 n where n = # ■ Consider binary numbers elements in set ■ E.g. 0101 S is the set {a, b, c}, what are all the subsets of S ? • ■ Let every bit position {1,…,n} be an item ■ { } – the empty set ■ Position i is in the set if bit i is 1 ■ {a}, {b}, {c} – one element sets ■ Position i is not in the set if bit i is 0 ■ {a, b}, {a, c}, {b, c} – two element sets ■ {a, b, c} – the original set ■ What is the set of all possible n-bit numbers? and hence the power set of S has 2 3 = 8 elements: ■ The powerset of {1,…n}. {{}, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}} CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 23 24

Example (contd.) Why sets? ■ n= 4, i.e, 4 bits, each representing 1 item ■ Programming - Recall a class … it is the set of all its possible objects. 1 2 3 4 ■ We can restrict the type of an object, which is {}, No item present 0 0 0 0 the set of values it can hold. {1}, Item 1 present ■ Example: Data Types 1 0 0 0 {2}, Item 2 present int set of integers (finite) 0 1 0 0 …… char set of characters (finite) {1, 2}, Items 1, 2 present 1 1 0 0 …… ■ Is N the same as the set of integers in a computer? {1, 2, 3}, Items 1, 2, 3 present ■ Is Q or R the same as the set of doubles in a 1 1 1 0 …… computer? {1, 2, 3, 4}, All items present 1 1 1 1 CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 25 26 Order Matters Tuples ■ What if order matters? ■ Order matters ■ Sets disregard ordering of elements ■ Duplicates matter ■ If order is important, we use tuples ■ Represented with parens ( ) ■ If order matters, then are duplicates ■ Examples important too? ■ (1, 2, 3) ≠ (3, 2, 1) ≠ (1, 1, 1, 2, 3, 3) CS 160, Summer Semester 2016 CS 160, Summer Semester 2016 27 28