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# Discrete Mathematics Exam 1 Study Guide 20123

TCU

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This 4 page Study Guide was uploaded by Rooshna Ali on Sunday February 21, 2016. The Study Guide belongs to 20123 at Texas Christian University taught by Susan Staples in Winter 2016. Since its upload, it has received 156 views. For similar materials see Discrete Mathematics I in Mathematics (M) at Texas Christian University.

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Date Created: 02/21/16

Exam 1 Study Guide (Sections 3-10) 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 0n Odd- an integer a is odd if there is an integer x such that a=2x+1 Prime- an integer p is prime if p1 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 1ba 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 n 1 choices for the first element, and for each one of those n choices there are n choices for the 1 2 second element, and each of these ordered pairs there are n3 choices, etc. the number of such lists is n1∗n 2n 3…∗n r Counting with and without repetition The number of lists of length k whose elements are chosen from a pool ofkn elements is n , if repetition is allowed n n−1 )(n−2 )…∗(n−k+1) , if no repetition is allowed This is called a permutation N factorial (n!) n!=n (n−1 )n−2 )…3∗2∗1 Summation Notation ( ∑) n ∑ f ( j) j=1 add all terms Product Notation ( ∏ ) n f ( j) j=1 multiply all terms Sets and Subsets 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 Element of a Set- an object is included in the set Notation: Cardinality of a statement- the number of elements in a set S Notation: |S| Empty set- the set which contains no objects Notation: 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 2. Show that every element in B belongs to A Template 6: Proving A B “Let x A … Therefore x B. We conclude A B. 1. Show A B 2. Show B A

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