Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

no text concepts found

Transcript

CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its elements The notation a ∈ A denotes that a is an element of the set A The notation a ∈ / A denotes that a is not an element of the set A The Roster Method of Describing a Set The syntax for the roster method is: S = {a, b, c, d} where the members of a set are delimited by curly braces and separated by commas The order does not matter: S = {a, b, c, d} = {b, c, a, d} Listing an element more than once does not change the set: S = {a, b, c, d} = {a, b, a, b, c, d, c} Ellipses (. . .) can be used when there is a clear pattern: S = {a, b, c, . . . , z} Set-Builder Notation Set-Builder notation is a method of describing sets where the properties that all members must meet are specified: S = {x | x > 0 ∧ x < 100} Reading set-builder notation: such that defined as S |{z} name of set z}|{ = { x |{z} all elements z}|{ | P(x) } | {z } P(x) is true Interval Notation [a, b] = {x | a ≤ x ≤ b} (closed interval) [a, b) = {x | a ≤ x < b} (a, b] = {x | a < x ≤ b} (a, b) = {x | a < x < b} (open interval) Some Important Sets N: the set of natural numbers Z: the set of integers Z+ : the set of positive integers R: the set of real numbers R+ : the set of positive real numbers C: the set of complex numbers Q: the set of rational numbers The Universal Set and Empty Set The universal set, U, is the set containing everything currently under consideration The empty set, denoted as ∅ or {}, is the set with no elements Note: the empty set is different than the set containing the empty set: ∅= 6 {∅} Set Equality Definition: Two sets are equal if and only if they have the same elements That is, the sets A and B are equal if and only if ∀x(x ∈ A ↔ x ∈ B) Subsets Definition: The set A is a subset of B if and only if every element of A is also an element of B The notation A ⊆ B is used to indicate that A is a subset of B A ⊆ B holds if and only if ∀x(x ∈ A → x ∈ B) Because a ∈ ∅ is always false, ∅ ⊆ S, for every set S Because a ∈ S → a ∈ S, S ⊆ S, for every set S Showing Things About Subsets To show that A ⊆ B, show that if x ∈ A, then x ∈ B To show that A 6⊆ B, find an element x ∈ A with x 6∈ B Set Equality with Subset Notation Recall A = B iff ∀x(x ∈ A ↔ x ∈ B) Using logical equivalences, we have A = B iff ∀x[(x ∈ A → x ∈ B) ∧ (x ∈ B → x ∈ A)] This is equivalent to: A⊆B ∧B ⊆A Proper Subsets Definition: If A ⊆ B, but A 6= B, then we say A is a proper subset of B, denoted by A ⊂ B If A ⊂ B, then ∀x(x ∈ A → x ∈ B) ∧ ∃x(x ∈ B ∧ x 6∈ A) is true Set Cardinality Definition: If there are exactly n distinct elements in the set S where n is a nonnegative integer, we say that S is finite otherwise S is infinite Definition: The cardinality of a finite set A, denoted by |A|, is the number of (distinct) elements in A Power Sets Definition: The set of all subsets of a set A, denoted P(A), is called the power set of A Example: If A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}} If a set has n elements, then the cardinality of the power set is 2n Tuples The ordered n-tuple, denoted as (a1 , a2 , . . . , an ) is an ordered collection the has a1 as its first element, a2 as its second element and so until an as its last element Two n-tuples are equal if and only if their corresponding elements are equal 2-tuples are called ordered pairs The ordered pairs (a, b) and (c, d) are equal iff a = c and b=d Cartesian Product Definition: The Cartesian Product of two sets A and B, denoted by A × B is the set of ordered pairs (a, b) where a ∈ A and b ∈ B That is, A × B = {(a, b) | a ∈ A ∧ b ∈ B} Example: A = {a, b}, B = {1, 2, 3} A × B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)} Cartesian Product Definition: The Cartesian products of the sets A1 , A2 , . . . , An , denoted by A1 × A2 × . . . × An , is the set of ordered n-tuples (a1 , a2 , . . . , an ) where ai ∈ Ai for i = 1, . . . n A1 × A2 × . . . × An = {(a1 , a2 , . . . , an ) | ai ∈ Ai for i = 1, 2, . . . , n} Truth Sets of Quantifiers Given a predicate P and a domain D, we define the truth set of P to be the set of elements in D for which P(x) is true The truth set is denoted by {x ∈ D | P(x)} Example: the truth set of P(x) where the domain is the integers and P(x) is “|x| = 1” is the set {−1, 1} Boolean Algebra Propositional logic and set theory are both instances of Boolean Algebra The operators in set theory are analogous to the corresponding operator in propositional logic All sets are assumed to be subsets of the universal set U Union Definition: Let A and B be sets. The union of the sets A and B, denoted by A ∪ B, is the set {x | x ∈ A ∨ x ∈ B} Intersection Definition: The union of sets A and B, denoted by A ∩ B, is the set {x | x ∈ A ∧ x ∈ B} Note: if the intersection is empty, then A and B are said to be disjoint Complement Definition: If A is a set, then the complement of A (with respect to U), denoted by A, is the set {x ∈ U | x ∈ 6 A} Difference Definition: Let A and B be sets. The difference of A and B, denoted by A − B, is the set {x | x ∈ A ∧ x 6∈ B} = A ∩ B Cardinality of the Union Operator Inclusion-Exclusion: |A ∪ B| = |A| + |B| − |A ∩ B| Symmetric Difference Definition: The symmetric difference of A and B, denoted A ⊕ B, is the set (A − B) ∪ (B − A) Set Identities Identity Laws: Domination Laws: Idempotent Laws: Complementation Law: Commutative Laws: Associative Laws: Distributive Laws: De Morgan’s Laws: Absorption Laws: Complement Laws: A ∪ ∅ = A, A ∩ U = A A ∪ U = U, A ∩ ∅ = ∅ A ∪ A = A, A ∩ A = A (A) = A A ∪ B = B ∪ A, A ∩ B = B ∩ A A ∪ (B ∪ C ) = (A ∪ B) ∪ C A ∩ (B ∩ C ) = (A ∩ B) ∩ C A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ) A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C ) A ∪ B = A ∩ B, A ∩ B = A ∪ B A ∪ (A ∩ B) = A, A ∩ (A ∪ B) = A A ∪ A = U, A ∩ A∅ Proving Set Identities Prove that each side of the identity is a subset of the other Use set builder notation and propositional logic Membership tables Proof of Second De Morgan Law Show that 1 A ∩ B ⊆ A ∪ B and 2 A∪B ⊆A∩B Proof of Second De Morgan Law First part: Show that A ∩ B ⊆ A ∪ B x ∈A∩B assumption x 6∈ A ∩ B def. of complement ¬((x ∈ A) ∧ (x ∈ B)) def. of intersection ¬(x ∈ A) ∨ ¬(x ∈ B) De Morgan’s Law in propositional logic x 6∈ A ∨ x 6∈ B def. of negation def. of complement x ∈A∨x ∈B x ∈A∪B def. of union Proof of Second De Morgan Law Second part: Show that x ∈A∪B (x ∈ A) ∨ (x ∈ B) (x 6∈ A) ∨ (x 6∈ B) ¬(x ∈ A) ∨ ¬(x ∈ B) ¬((x ∈ A) ∧ (x ∈ B)) ¬(x ∈ A ∩ B) x ∈A∩B A∪B ⊆A∩B assumption def. union def. of complement def. of negation De Morgan’s Law in propositional logic def. of intersection def. of complement Proof of Second De Morgan Law: Set-Builder Notation A∩B = = = = = = = = {x | x 6∈ A ∩ B} {x | ¬(x ∈ (A ∩ B))} {x | ¬(x ∈ A ∧ x ∈ B)} {x | ¬(x ∈ A) ∨ ¬(x ∈ B)} {x | x 6∈ A ∨ x 6∈ B} {x | x ∈ A ∨ x ∈ B} {x | x ∈ A ∪ B} A∪B def. of complement def of 6∈ def. of intersection De Morgan’s law def. of 6∈ def. of complement def. of union meaning of notation Membership Table Example Show A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C ) A 1 1 1 1 0 0 0 0 B 1 1 0 0 1 1 0 0 C 1 0 1 0 1 0 1 0 B ∩C 1 0 0 0 1 0 0 0 A ∪ (B ∩ C ) 1 1 1 1 1 0 0 0 A∪B 1 1 1 1 1 1 0 0 A∪C 1 1 1 1 1 0 1 0 (A ∪ B) ∩ (A ∪ C ) 1 1 1 1 1 0 0 0

Related documents