Q Integers Whole Numbers Natural Numbers Z W

Q Integers Whole Numbers Natural Numbers Z W N Type Subset Cardinality Uncountable Closure Properties +, -, ×, ÷ Notation +, -, ×, ÷ Countable Range Name Open Interval *Irrational numbers have none Domain, Codomain, and Range +, -, × All Numbers without fractional component Countable Closed Interval +, × All positive integers Countable +, × All positive integers less 0 Countable Subsets Ways to Write Sets Meaning Notation Way For sets with Example Remarks Every element of A is also an Finite elements {3, -3} [≠(3, -3)] Half-open Set Roster Order and repetition do not matter element of B Fixed Pattern {…, -2, -1, 0, 1, 2, …} interval (A is a subset of B) A⊆B General Form { x ∈ S | P(x) } X S * For any set A, A is a subset of A, A ⊆ A. Set Builder Proper Subset A ⊆ B but A ≠ B Anything {x ∈ R+| x<10} Variable Underlying set for x 1 ⊈ {{1}, {2, 3}, {4, 5}} 1 ∉ {{1}, {2, 3}, {4, 5}} {1} ⊈ {{1}, {2, 3}, {4, 5}} Cartesian Product {{1}} ∉ {{1}, {2, 3}, {4, 5}} {{1}} ⊆ {{1}, {2, 3}, {4, 5}} {1} ∈ {{1}, {2, 3}, {4, 5}} Equality of Functions Image and Pre-image All elements in codomain mapped from exactly one element in domain Not all elements in codomain must be mapped to e. g. {n | n is an even integer} Power Sets Set of all subsets of A |P (A)| = 2 |A| Proving/Disproving Notation P(x) Underlying condition in terms of x Open Sometimes the underlying set may be omitted fromray the notation. Examples 3 Types of Functions Definition Injective Function Rational Numbers Function Number Sets Meaning All numbers less complex All numbers that can be expressed as a quotient Partitions Closed ray Cartesian Distributive Laws Inverse Image Surjective Function Sets (Big to small) Symbol R Real Numbers Elements in codomain can be mapped from more than one element in domain All elements in codomain must be mapped to, and range = codomain Proving Subsets - Element Chasing Method Disproving Proving A ⊆ B A⊆B Notation Expanded Ways to count number of partitions of a set Set of elements in Identity and Characteristic Functions Indexed collection of sets Example Notation Function on Cartesian Products The domain and/or codomain of a function can be a Cartesian product of two or more sets Examples: Meaning Inverse Functions Disproving Set Relations - Find a counterexample of sets that disproves the relation Set Relations with Conditions Use ‘if’ to prove ‘then’ - ‘If’ condition is not always used at the start - Use element method, and use ‘if’ condition when necessary Indexed Collection Distributions Proving Set Identity – Element Method Proving A ⊆ B Composite Functions Set Logic Commutative Law Composition, Inverse, and Identity Associative Law Composition of Injective, Surjective, and Bijective Functions Prove A ⊆ B and B ⊆ A to prove A = B Set Operations (Biconditional) Prove that it is both injective and surjective All elements in codomain mapped To prove a set is infinite yet countable, show that all from exactly one element in domain elements in the set can be mapped to the set of positive integers using a bijective function All elements in the codomain are mapped to, and range = codomain Countable Sets Prove A ⊆ B and B ⊆ A to prove A = B Proving Set Identity – Algebra of Sets Proving A = B Use set logic to derive the result. Same as normal logic Bijective function Indexed collection of sets Distributive Law Attaching and Cancelling Functions MI and Well-Ordering Principal (MORE MI ON NEXT PAGE)

Congruence Modulo n Relation Greatest Common Divisor Remainder of Large Powers Find remainder before raising power Binary Sum Notation Let R be a relation from A to B. If a ∈ A is related to b ∈ B, we write a R b for short. If a is not related to b, we write a R b for short. Order Pair Representation Euclidian and Reverse Euclidian Algorithms Inverse Relation Congruence Classes and Relations Reducing Congruence Modulo Equivalence Relation Linear Combinations Modular Arithmetic Integers Modulo n Relatively Prime/Co-Prime Equivalence Classes + [0] [1] [2] [1] [2] [0] [1] Equivalence Relations and Partitions + [0] [1] [2] [3] [1] [2] [3] [0] [1] [3] [0] [1] [2] + [0] [1] [2] [3] [4] [1] [2] [3] [4] [0] [1] [3] [4] [0] [1] [2] [3] + [0] [1] [2] [3] [4] [5] [1] [2] [3] [4] [5] [0] [1] [3] [4] [5] [0] [1] [2] [3] [5] [0] [1] [2] [3] [4] Finding Inverse of a modulo n Using Inverse Modulo n Cancellation Theorem * [0] [1] [2] Logical Connectives Type Standard Form Symbolic Form ~P Negation not P P∧Q Conjunction P and/but/nor Q P∨Q Disjunction P or Q Truth Tables Negation Conjunction Disjunction P T F P F T Q T T T F F T Commutative Law F F Associative Law Conditional Statements Forms of Conditionals Truth Table P Q P→Q [1] [0] [1] [2] [0] [2] [1] * [0] [1] [2] [3] P∨c≡P [3] [0] [3] [2] [1] * [0] [1] [2] [3] [4] [0] [0] [0] [1] [2] [3] [4] Even integer Negation of Conditional ∧ (Q∨ → P → Q ≡ ~((P ≡ → Q) ~P Q P )) ~ (P → Q) ≡ ~(~P ∨ Q) ≡ → Q) ~(~P) ~Q ≡ ~(P ∨ ~(Q∧→ P) ∧ ~Q ≡ ≡ (P ∧ ~Q) ∨P (Q ∧ ~P) [3] [0] [3] [1] [4] [2] [4] [0] [4] [3] [2] [1] * [0] [1] [2] [3] [4] [5] [0] [0] [1] [2] [3] [4] [5] [2] [0] [2] [4] [3] [0] [3] [4] [0] [4] [2] [5] [0] [5] [4] [3] [2] Common Definitions [1] Fermat’s Little Theorem Definition Step Intro Basis Mathematical Induction Write Let P(n) be the statement: (statement) for (condition on n) Prove P(initial n), usually P(1), is true. Inductive Hypothesis Want to show Working Step Conclusion Intro Basis True/False - Double Quantified Statements Type [2] [0] [2] [4] [1] [3] Odd Integer Rational Number P(k): (statement in terms of k) for (condition on k) P(k+1): (statement in terms of k+1), assuming P(k) is true LHS of P(k+1) =. . . (use P(k) here). . . = RHS of P(k+1) METHOD NOT FIXED Strong Mathematical Induction Write Let P(n) be the statement: (statement) for (condition on n) Prove P(initial n’s) are true, where m* is the last initial n How ∃x ∃y, T Find values of x, y P(x, y) F Give a proof for general x, y P∧t≡P [2] [0] [2] (P → Q) ∧ (Q → P ) ∀y ∃x, T For a general y, find a value of x P(x, y) F Find a value of y for all x Identity Law [1] [0] [1] [2] [3] Name ≡ ∀x ∀y, T Give a proof for general x, y P(x, y) F Find values of x, y Distributive Law [0] [0] [0] Negation of Biconditional Q when(ever) P P only if Q T T T If P then Q Q if P Q is necessary for P T F F Biconditional Statements sufficient for Q F T T P implies Q PPis↔ Q P↔Q F F T P iff Q P Q T T T P if and only if Q T F F P and Q are equivalent F T T T T F F T P is necessary and suffici for Q F T Logical Equivalences F F T T P∧Q≡Q∧P P∨Q≡Q∨P F (P ∧ Q) F ∧R F ≡ P F∧ (Q ∧ R) (P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R) P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R) P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R) [0] [0] ∃x ∀y, T Find a value of x for all y P(x, y) F For a general x, find a value of y Inductive Quotient Remainder Thm Hypothesis Prime P(k): (statement in terms of k) for (condition on k) Want to show LHS of P(k+1) =. . . (use P(k) here). . . = RHS of P(k+1) METHOD NOT FIXED