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Discrete Math Final Exam Study Guide

by: Rooshna Ali

Discrete Math Final Exam Study Guide 20123

Marketplace > Texas Christian University > Mathematics (M) > 20123 > Discrete Math Final Exam Study Guide
Rooshna Ali

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About this Document

This study guide will prepare you for the final exam
Discrete Mathematics I
Susan Staples
Study Guide
Math, Discrete math, Discrete Mathematics, Discrete, Mathematics
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This 7 page Study Guide was uploaded by Rooshna Ali on Monday May 9, 2016. The Study Guide belongs to 20123 at Texas Christian University taught by Susan Staples in Winter 2016. Since its upload, it has received 33 views. For similar materials see Discrete Mathematics I in Mathematics (M) at Texas Christian University.


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Date Created: 05/09/16
Final Exam Study Guide (Sections 3-12, 19-22, Chapter 5, Sections 35-37) Basic Definitions • Integers o Notation Z o Z = {… -3, -2, -1, 0, 1, 2, 3} • Rational Numbers o Notation Q o Any number that can be represented in a fraction • Natural Numbers o Notation N o N = {0, 1, 2, 3, …} • Even- if it is divisible by 2 • Divisible- Let a and b be integers o a is divisible by b provided that there is an integer c such that b ⋅ c = a o “a is divisible by b” § b divides a § b is a factor of a § b is a divisor of a o notation: b|a § b divides a o Note: 2|0 but 0∤2 because 2 ≠ 0⋅n • Odd- an integer a is odd if there is an integer x such that a=2x+1 • Prime- an integer p is prime if p>1 and the only positive divisors of p are p and 1 • Composite- a positive integer a is composite, provided there is an integer b with 1<b<a for which b|a Logical Statements and Truth Evaluations • Statement- a declarative sentence that can be assessed for its correctness or validity (truth) or its invalidity (false) • ‘And’ statement ( ) A B A B T T T T F F F T F F F F • ‘Or’ statement ( ) A B A B T T T T F T F T T F F F • ‘Not’ statement ( ¬) A ¬ A T F F T • ‘If-then’ statement (⇒) A B A B T T T T F F F T T F F T • ‘If-and-only-if’ (“iff”) statement (⇔) A B A ⇔ B T T T T F F F T F F F T • Theorem- a statement known to be true, it can be proven • Conjecture- a statement thought to be true, but it has not yet been proven • Vacuous Truth- statements in the form “If A, then B” in which the hypothesis A never happens (i.e. it’s impossible), because there are no instances where the statement is false • Counterexample- an example that shows a theorem or result is false • Tautology- a compound logic statement that is true no matter what truth values are assigned to the variables • Fallacy (Contradiction)- a compound logic statement that is false no matter what truth values are assigned to the variables • Contingency- a statement that is sometimes false and sometimes true • Logical Equivalence- if two compound logic statements have the same truth values for all possible assignments of truth values to the variables • Variations of the Conditional The conditional p ⇒ q § Converse: q ⇒ p § Inverse: ¬p ⇒ ¬ q § Contrapositive: ¬ q ⇒ ¬ p *Note: The conditional and contrapositive are logically equivalent* Numerical problems and counting (Sections 8 and 9) • List- an ordered sequence of objects • Generalized Multiplication Principle of Counting- consider creating a list of length r. Suppose there are ???? ▯hoices for the first element, and for each one of those ???? ▯choices there are ???? choices for the second element, and each of these ordered pairs there are ▯ ????▯ choices, etc. à the number of such lists is ???? ▯ ???? ▯ ???? ∗▯…∗ ???? ▯ • Counting with and without repetition The number of lists of length k whose elements are chosen from a pool of n elements is § ???? , if repetition is allowed § ???? ???? − 1 ???? − 2 …∗ (???? − ???? + 1), if no repetition is allowed • This is called a permutation • N factorial (n!) ????! = ???? ???? − 1 ???? − 2 …3 ∗ 2 ∗ 1 • Summation Notation ( ∑ ) ▯▯▯????(????) à add all terms • Product Notation ( ∏ ) ▯▯▯ ????(????) à multiply all terms Sets (Sections 10 & 12) • Set- A repetition: free unordered collection of objects • Subset- If we have two sets, A and B. We say A is a subset of B, provided every element of A is in B Notation: A ⊆ B • Proper subset- If we have two sets, A and B. We say A is a subset of B, provided not every element of A is in B (Can’t be exact same) Notation: A ⊂ B • Element of a set- when an object is in a set Notation: ∈ • Cardinality of a set- the number of of element in a set S Notation: ???? • Empty set- the set that contains no objects Notation: ∅ • Power set (abbreviated as P(A) as power of A) Ex. Suppose A = {a,b,c} P(A) = { { }, {a}, {b}, {c}, {a,b}, {a,c}, {b,c} {a,b,c} } Note: See that ???? = 3 and ????(????) = 8 ** if ???? < ∞, then ????(????) = 2 ▯ • Intersection- the set of all elements that are in both A and B Notation: A ∩ B • Union- the set of all elements that are are in A or B (or both) Notation: A ∪ B • Complement of sets- Let U be the set of all elements in the universe of discourse We define the complement of A (written as ????) as the set of all elements that are in U but not in A That is ???? = ???? − ???? • Cartesian product of sets (of sets A and B—denoted as A x B)- the set of all ordered pairs formed by taking an element from A together from B • Disjoint sets- Let A and B be sets A ∩ B = ∅ Counting Problems for sets (Sections 12 & 19) • The Addition Rule for Mutually Disjoint Sets: |A|+|B|+|C| = |A ∪ B ∪ C| • The Difference Rule for sets A and B with⊆ A: |A|-|B| = |A|-|B| • The Inclusion-Exclusion rules for the two and three set cases: Two cases: |A ∪ B| = |A|+|B| - |A ∩ B| Three cases: |A ∪ B ∪ C| = |A|+|B|+|C| - |A ∩ B|- |A ∩ C| - |B ∩ C|+ |A ∩ B ∩ C| Universal and Existential Quantifiers (Section 11) • Predicate: states a property a variable “x” has Notation: P(x)…. For x satisfies predicate P • Universe of Discourse: (for a variable x) Specifies the values of x under consideration • Universal Quantification: (of the P(x) is the statement) “For all x in the universe of discourse, P(x) is true” Notation: ∀xP(x) Synonyms: for all, for each, every, given any, etc. • Existential Quantification: (of the P(x) is the statement) “There exists an x in the universal discourse for which P(x) is true” Notation: ∃xP(x) Synonyms: There exists, there is, for some, for at least one, etc. • Negation of Quantified Statements: o ¬ (∀xP(x)) ≡ ∃x (¬ P(x)) o ¬ (∃xP(x))≡ ∀x (¬ P(x)) o ¬ (P(x) Q(x)) ≡ (¬ (P(x) Q(x))) o ¬ (P(x) Q(x))≡ ¬ (P(x) Q(x))) o ¬ (P(x) ⇒ Q(x))≡ (P(x) Q(x)) Chapter 5: Functions Let A and B be sets • Function: (a function f from set A to B) associates with each element a ∈ A a unique element of b ∈ B for which f(a)=b • Image of a function: here in “f(a)=b”, b is the image of a • Maps: we say f maps a to f(a) Notation for this function is f: AàB • Domain: call A the domain of the function • Codomain: call B the codomain of the function • Range: the set of all images of elements of A under f • One-to-one function: a function f: AàB is called one-to-one, if f(s) = f(t) implies s=t, for all s, t ∈ A • Onto function: a function f: AàB is called onto if every element of b is the image of some element of A under f. That is, f is onto if for any b ∈ B, there exists an a ∈ A for which f(a)=b • Bijection (a.k.a. one-to-one correspondence): a function f: AàB is called bijection if f is both one-to-one and onto • Composition of functions: Let g: AàB be a function from A to B Let f: BàC be a function from B to C Then the composition of f °g is the function from A to C defined by f ° g(a) = f(g(a)) • Inverse functions: Let f be a bijection from A to B, that is f: AàB is onto and one-to-one Then we can define the inverse function f : BàA by f (b) = a, where a is the unique element in A for which f(a)=b Proof Templates • Template 1: Direct proof of “if-then” theorem (a.k.a. “The Forwards-Backwards Method”) Prove if A, then B 1. Restate A 2. Restate B 3. Go forwards/backwards until “AHA!” 4. Organize proof • Template 2: “If-and-only-if” proofs Prove A ⇔ B 1. Prove A ⇒ B 2. Prove B ⇒ A • Template 3: Disprove an “if-then” statement 1. Find a counterexample • Template 4: Proving logical equivalence 1. Create truth table for each statement 2. Confirm the two statements agree in the truth value for every case • Template 5: Proving set A = set B 1. Show that every element in A belongs to B (show A ⊆ B) 2. Show that every element in B belongs to A (show B ⊆ A) • Template 6: Proving A ⊆ B “Let x ∈ A … Therefore x ∈ B. We conclude A ⊆ B. • Template 11: Proof by contrapositive 1. In order to prove A ⇒ B 2. Prove ¬B ⇒ ¬A a. So assume not B b. Work to show not A **Clues when to use- “Not B” offers a more natural starting point than “A” If “B” contains an OR statement à cuz “not B” will then contain an “AND” statement • Template 12: Proof by contradiction 1. Assume A 2. Assume not B 3. Proceed step-by-step until you reach a contradiction • Template 17: Math Induction Setting P(n) is a statement that is defined for all integers n ≥ a, where a is a fixed natural number 1. “Basic Step” Prove special case P(n) is true 2. “Inductive Step” Show for all integers k ≥ a, if P(a) is true then P(k+1) is true • Template 20: Prove a function is one-to-one Direct Method: For any s, t ∈ A, assume f(s) = f(t) : : (algebra most likely) : Therefore, s = t • Template 21: Prove a function is onto To prove f: AàB is onto, take any b ∈ B : : (work the algebra/definition f “backwards” to find a) : Conclude for some a ∈ A, that f(a)=b


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