Gibbins, Nicholas

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Logic for Web ScientistsNicholas GibbinsFebruary 27, 20141 SetsDefinition: SetA set is an unordered collection of objects, without duplicates. A set A containing theobjects a, b and c is written as A = {a, b, c}.Definition: Empty SetThe empty set (written ∅ or {}) is the set containing nothing.Definition: Set MembershipAn object a is a member of a set A (written a ∈ A) if it is contained within that collection.Note: a ∈ A can be read as “a is a member of A” or “a belongs to A”.Definition: Set EqualityTwo sets which contain the same objects are considered to be equal (the order of theobjects is unimportant).Example: If A = {a, b, c}, B = {b, a, c} and C = {a, b, d}, then A = B but A 6= CDefinition: CardinalityThe cardinality of a set A (written |A| or #A) is the number of members of A.Example: If A = {a, b, c}, then |A| = 3Definition: SubsetA set A is a subset of a set B (written A ⊆ B) if every member of A is also a member of B.Example: If A = {a, b}, B = {a, b, c} and C = {a, c, d}, then A ⊆ B, but A 6⊆ C1Definition: Strict SubsetA set A is a strict subset of a set B (written A ⊂ B) if every member of A is also amember of B, and A 6= B.Example: If A = {a, b}, B = {a, b, c} and C = {a, b}, then A ⊂ B, but A 6⊂ CDefinition: Set IntersectionThe intersection of two sets A and B (written as A ∩ B) is the set containing everyobject which is both a member of A and a member of B.Example: If A = {a, b, c} and B = {a, c, d} then A ∩B = {a, c}Definition: Set UnionThe union of two sets A and B (written as A∪B) is the set containing every object thatis a member of A or a member of B, or a member of both A and B.Example: If A = {a, b, c} and B = {a, c, d} then A ∪B = {a, b, c, d}Mnemonic: ∪ stands for U(nion)Definition: Set DifferenceThe difference of two sets A and B (written as A−B) is the set of every object that isa member of A but not a member of B.Example: If A = {a, b, c} and B = {a, c, d} then A−B = {b}Note: (A−B) 6= (B −A)Definition: PowersetThe powerset of A (written P(A) or 2A) is the set containing all possible subsets of A,including A and the empty set.Example: If A = {a, b, c}, thenP(A) = {{a, b, c}, {a, b}, {a, c}, {b, c}, {a}, {b}, {c}, {}}Note: |P(A)| = 2|A|.2Definition: Set ComprehensionRather than explicitly list the members of a set as A = {a1, . . . , an} ,we can define a setby specifying the properties that its members must have. This is known as set compre-hension.Set comprehension is expressed using set-builder notation, for which the general form is{x : φ(x)}, where x is a variable and φ(x) is a predicate containing x which holds truefor all members of the set. {x : φ(x)} can be read as “the set of x for which φ(x) is true”.Example: {x : x ∈ Z ∧ x > 0}The set of positive integers - Z is the set of integers.Read as: “the set of x’s where x is an integer and x is greater than zero”.Example: {x : x ∈ Z ∧ x = x2}The set of integers which are equal to their square: {0, 1}Example: {〈x, y〉 : x ∈ A ∧ y ∈ B}The set of pairs 〈x, y〉 where x is a member of set A and y is a member of set B. Thisis the definition of the Cartesian product A×B using set-builder notation.Definition: TupleA tuple is an ordered collection of objects, which may include duplicates. The tuplecontaining a, b, c and a, in that order, is written 〈a, b, c, a〉Definition: ArityThe degree or arity of a tuple is the number of objects in the tuple.Definition: PairA tuple containing two objects (a tuple of arity 2) is known as a pair.Definition: Cartesian ProductThe Cartesian product of two sets A and B (written A × B) is a set of pairs, whereeach pair contains one member from A and one member from B, and which contains allpossible combinations of members from A and B.Example: If A = {a, b, c} and B = {c, d, e}, thenA×B = {〈a, c〉, 〈a, d〉, 〈a, e〉, 〈b, c〉, 〈b, d〉, 〈b, e〉, 〈c, c〉, 〈c, d〉, 〈c, e〉}Note: |A×B| = |A| ∗ |B|3Definition: Binary RelationA binary relation R from set A to set B is a set of pairs, where each pair contains onemember from A and one member from B.Example: If A = {a, b, c} and B = {c, d, e}, then a possible relation R from A to Bmight be:R = {〈a, c〉, 〈a, e〉, 〈b, d〉, 〈c, c〉, 〈c, d〉, }Note: R ⊆ A×BDefinition: DomainThe domain of a binary relation R is the set that the relation goes from.Example: The domain of R in the above example is A.Definition: RangeThe range of a binary relation R is the set that the relation goes to.Example: The range of R in the above example is B.Mnemonic: the range of a cannon is the distance to which it can fire a cannonball.2 LogicDefinition: PredicateA predicate is a truth-valued expression. That is, a predicate can either be true or false.Example: “a ∈ A” is a predicate (either a is a member of A, in which case “a ∈ A”is true, or a is not a member of A, in which case “a ∈ A” is false). “A × B” is not apredicate, because its value is a set.Definition: Logical OperatorsPredicates may be combined to form compound predicates by using the logical operators:conjunction (∧), disjunction (∨), negation (¬) and implication (⇒) .Definition: Conjunction (logical and)The conjunction of two predicates φ and ψ (written as φ∧ψ, and read as “φ and ψ”) istrue if both φ and ψ are true.4Mnemonic: ∧ stands for A(nd)φ ψ φ ∧ ψfalse false falsefalse true falsetrue false falsetrue true trueDefinition: Disjunction (logical or)The disjunction of two predicates φ and ψ (written as φ ∨ ψ, and read as “φ or ψ”) istrue if either φ is true or ψ is true (or if both φ and ψ are true – ∨ is the inclusive-or).φ ψ φ ∨ ψfalse false falsefalse true truetrue false truetrue true trueDefinition: NegationThe negation of a predicate φ (written as ¬φ, and read as “not φ”) is true if φ is false.φ ¬φfalse truetrue falseDefinition: Implicationφ ψ φ⇒ ψfalse false truefalse true truetrue false falsetrue true true5

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