THEOREMS FROM GRIES AND SCHNEIDER’S LADM
J. STANLEY WARFORD
Summary. It is a assortment of the axioms and theorems in Gries and Schneider’s e-book A Logical Method to Discrete Math (LADM), Springer-Verlag, 1993. The numbering is according to that textual content. Extra theorems not included or numbered in LADM are indi- cated by a three-part quantity. This doc serves as a reference for homework workouts and taking exams.
Desk of Precedences (a) [x := e] (textual substitution) (highest priority) (b) . (operate software) (c) unary prefix operators: + − ¬ # ∼ P (d) ∗∗ (e) · / ÷ mod gcd (f) + − ∪ ∩ × ◦ • (g) ↓ ↑ (h) # (i) ◁ ▷ ^ (j) = < > ∈ ⊂ ⊆ ⊃ ⊇ | (conjunctional) (ok) ∨ ∧ (l) ⇒ ⇐ (m) ≡ (lowest priority)
All nonassociative binary infix operators affiliate from left to proper besides ∗∗, ◁, and ⇒, which affiliate from proper to left.
Definition of /: The operators on traces (j), (l), and (m) could have a slash / by means of them to indicate negation—e.g. x /∈ T is an abbreviation for ¬(x ∈ T ).
Some Primary Varieties Identify Image Sort (set of values) integer Z integers: …,−three,−2,−1,zero,1,2,three,… nat N pure numbers: zero,1,2,… constructive Z+ constructive integers: 1,2,three,… damaging Z− damaging integers: −1,−2,−three,… rational Q rational numbers: i/ j for i, j integers, j ̸= zero reals R actual numbers constructive reals R+ constructive actual numbers bool B booleans: true, f alse
Date: January eight, 2020. 1
2 J. STANLEY WARFORD
Theorems of the Propositional Calculus Equivalence and true. (three.1) Axiom, Associativity of ≡ : ((p ≡ q) ≡ r) ≡ (p ≡ (q ≡ r)) (three.2) Axiom, Symmetry of ≡ : p ≡ q ≡ q ≡ p (three.three) Axiom, Id of ≡ : true ≡ q ≡ q (three.four) true (three.5) Reflexivity of ≡ : p ≡ p
Negation, inequivalence, and f alse. (three.eight) Definition of f alse : f alse ≡ ¬true (three.9) Axiom, Distributivity of ¬ over ≡ : ¬(p ≡ q) ≡ ¬p ≡ q (three.10) Definition of ̸≡ : (p ̸≡ q) ≡ ¬(p ≡ q) (three.11) ¬p ≡ q ≡ p ≡ ¬q (three.12) Double negation: ¬¬p ≡ p (three.13) Negation of f alse: ¬ f alse ≡ true (three.14) (p ̸≡ q) ≡ ¬p ≡ q (three.15) ¬p ≡ p ≡ f alse (three.16) Symmetry of ̸≡ : (p ̸≡ q) ≡ (q ̸≡ p) (three.17) Associativity of ̸≡ : ((p ̸≡ q) ̸≡ r) ≡ (p ̸≡ (q ̸≡ r)) (three.18) Mutual associativity: ((p ̸≡ q) ≡ r) ≡ (p ̸≡ (q ≡ r)) (three.19) Mutual interchangeability: p ̸≡ q ≡ r ≡ p ≡ q ̸≡ r (three.19.1) p ̸≡ p ̸≡ q ≡ q
Disjunction. (three.24) Axiom, Symmetry of ∨ : p ∨ q ≡ q ∨ p (three.25) Axiom, Associativity of ∨ : (p ∨ q)∨ r ≡ p ∨(q ∨ r) (three.26) Axiom, Idempotency of ∨ : p ∨ p ≡ p (three.27) Axiom, Distributivity of ∨ over ≡ : p ∨(q ≡ r) ≡ p ∨ q ≡ p ∨ r (three.28) Axiom, Excluded center: p ∨¬p (three.29) Zero of ∨ : p ∨ true ≡ true (three.30) Id of ∨ : p ∨ f alse ≡ p (three.31) Distributivity of ∨ over ∨ : p ∨(q ∨ r) ≡ (p ∨ q)∨(p ∨ r) (three.32) p ∨ q ≡ p ∨¬q ≡ p
Conjunction. (three.35) Axiom, Golden rule: p ∧ q ≡ p ≡ q ≡ p ∨ q (three.36) Symmetry of ∧ : p ∧ q ≡ q ∧ p (three.37) Associativity of ∧ : (p ∧ q)∧ r ≡ p ∧(q ∧ r) (three.38) Idempotency of ∧ : p ∧ p ≡ p (three.39) Id of ∧ : p ∧ true ≡ p (three.40) Zero of ∧ : p ∧ f alse ≡ f alse
THEOREMS FROM LADM three
(three.41) Distributivity of ∧ over ∧ : p ∧(q ∧ r) ≡ (p ∧ q)∧(p ∧ r) (three.42) Contradiction: p ∧¬p ≡ f alse (three.43) Absorption:
(a) p ∧(p ∨ q) ≡ p (b) p ∨(p ∧ q) ≡ p
(three.44) Absorption: (a) p ∧(¬p ∨ q) ≡ p ∧ q (b) p ∨(¬p ∧ q) ≡ p ∨ q
(three.45) Distributivity of ∨ over ∧ : p ∨(q ∧ r) ≡ (p ∨ q)∧(p ∨ r) (three.46) Distributivity of ∧ over ∨ : p ∧(q ∨ r) ≡ (p ∧ q)∨(p ∧ r) (three.46.1) Consensus: (p ∧ q)∨(¬p ∧ r)∨(q ∧ r) ≡ (p ∧ q)∨(¬p ∧ r) (three.47) De Morgan:
(a) ¬(p ∧ q) ≡ ¬p ∨¬q (b) ¬(p ∨ q) ≡ ¬p ∧¬q
(three.48) p ∧ q ≡ p ∧¬q ≡ ¬p (three.49) p ∧(q ≡ r) ≡ p ∧ q ≡ p ∧ r ≡ p (three.50) p ∧(q ≡ p) ≡ p ∧ q (three.51) Substitute: (p ≡ q)∧(r ≡ p) ≡ (p ≡ q)∧(r ≡ q) (three.52) Equivalence: p ≡ q ≡ (p ∧ q)∨(¬p ∧¬q) (three.53) Unique or: p ̸≡ q ≡ (¬p ∧ q)∨(p ∧¬q) (three.55) (p ∧ q)∧ r ≡ p ≡ q ≡ r ≡ p ∨ q ≡ q ∨ r ≡ r ∨ p ≡ p ∨ q ∨ r
Implication. (three.57) Definition of Implication: p ⇒ q ≡ p ∨ q ≡ q (three.58) Axiom, Consequence: p ⇐ q ≡ q ⇒ p (three.59) Implication: p ⇒ q ≡ ¬p ∨ q (three.60) Implication: p ⇒ q ≡ p ∧ q ≡ p (three.61) Contrapositive: p ⇒ q ≡ ¬q ⇒ ¬p (three.62) p ⇒ (q ≡ r) ≡ p ∧ q ≡ p ∧ r (three.63) Distributivity of ⇒ over ≡ : p ⇒ (q ≡ r) ≡ (p ⇒ q) ≡ (p ⇒ r) (three.63.1) Distributivity of ⇒ over ∧ : p ⇒ q ∧ r ≡ (p ⇒ q)∧(p ⇒ r) (three.63.2) Distributivity of ⇒ over ∨ : p ⇒ q ∨ r ≡ (p ⇒ q)∨(p ⇒ r) (three.64) p ⇒ (q ⇒ r) ≡ (p ⇒ q) ⇒ (p ⇒ r) (three.65) Shunting: p ∧ q ⇒ r ≡ p ⇒ (q ⇒ r) (three.66) p ∧(p ⇒ q) ≡ p ∧ q (three.67) p ∧(q ⇒ p) ≡ p (three.68) p ∨(p ⇒ q) ≡ true (three.69) p ∨(q ⇒ p) ≡ q ⇒ p (three.70) p ∨ q ⇒ p ∧ q ≡ p ≡ q (three.71) Reflexivity of ⇒ : p ⇒ p (three.72) Proper zero of ⇒ : p ⇒ true ≡ true (three.73) Left identification of ⇒ : true ⇒ p ≡ p
four J. STANLEY WARFORD
(three.74) p ⇒ f alse ≡ ¬p (three.74.1) ¬p ⇒ f alse ≡ p (three.74.2) p ⇒ ¬p ≡ ¬p (three.75) f alse ⇒ p ≡ true (three.76) Weakening/strengthening:
(a) p ⇒ p ∨ q (Weakening the ensuing) (b) p ∧ q ⇒ p (Strengthening the antecedent) (c) p ∧ q ⇒ p ∨ q (Weakening/strengthening) (d) p ∨(q ∧ r) ⇒ p ∨ q (e) p ∧ q ⇒ p ∧(q ∨ r)
(three.76.1) p ∧ q ⇒ p ∨ r (Weakening/strengthening) (three.76.2) (p ⇒ q) ⇒ ((q ⇒ r) ⇒ (p ⇒ r)) (three.77) Modus ponens: p ∧(p ⇒ q) ⇒ q (three.77.1) Modus tollens: (p ⇒ q)∧¬q ⇒ ¬p (three.78) (p ⇒ r)∧(q ⇒ r) ≡ p ∨ q ⇒ r (three.78.1) (p ⇒ r)∨(q ⇒ r) ≡ p ∧ q ⇒ r (three.79) (p ⇒ r)∧(¬p ⇒ r) ≡ r (three.80) Mutual implication: (p ⇒ q)∧(q ⇒ p) ≡ (p ≡ q) (three.81) Antisymmetry: (p ⇒ q)∧(q ⇒ p) ⇒ (p ≡ q) (three.82) Transitivity:
(a) (p ⇒ q)∧(q ⇒ r) ⇒ (p ⇒ r) (b) (p ≡ q)∧(q ⇒ r) ⇒ (p ⇒ r) (c) (p ⇒ q)∧(q ≡ r) ⇒ (p ⇒ r)
(three.82.1) Transitivity of ≡ : (p ≡ q)∧(q ≡ r) ⇒ (p ≡ r) (three.82.2) (p ≡ q) ⇒ (p ⇒ q) Leibniz as an axiom. This part makes use of the next notation: E zX means E[z := X]. (three.83) Axiom, Leibniz: e = f ⇒ E ze = E zf (three.84) Substitution:
(a) (e = f )∧ E ze ≡ (e = f )∧ E zf (b) (e = f ) ⇒ E ze ≡ (e = f ) ⇒ E zf (c) q ∧(e = f ) ⇒ E ze ≡ q ∧(e = f ) ⇒ E zf
(three.85) Change by true: (a) p ⇒ E zp ≡ p ⇒ E ztrue (b) q ∧ p ⇒ E zp ≡ q ∧ p ⇒ E ztrue
(three.86) Change by f alse: (a) E zp ⇒ p ≡ E zf alse ⇒ p (b) E zp ⇒ p ∨ q ≡ E zf alse ⇒ p ∨ q
(three.87) Change by true: p ∧ E zp ≡ p ∧ E ztrue (three.88) Change by f alse: p ∨ E zp ≡ p ∨ E zf alse (three.89) Shannon: E zp ≡ (p ∧ E ztrue)∨(¬p ∧ E zf alse) (three.89.1) E ztrue ∧ E zf alse ⇒ E
z p
THEOREMS FROM LADM 5
Extra theorems regarding implication. (four.1) p ⇒ (q ⇒ p) (four.2) Monotonicity of ∨ : (p ⇒ q) ⇒ (p ∨ r ⇒ q ∨ r) (four.three) Monotonicity of ∧ : (p ⇒ q) ⇒ (p ∧ r ⇒ q ∧ r)
Proof method metatheorems. (four.four) Deduction (assume conjuncts of antecedent):
To show P1 ∧ P2 ⇒ Q, assume P1 and P2, and show Q. You can’t use textual substitution in P1 or P2.
(four.5) Case Assessment: If E ztrue and E zf alse are theorems, then so is E z P.
(four.6) Case Assessment: (p ∨ q ∨ r)∧(p ⇒ s)∧(q ⇒ s)∧(r ⇒ s) ⇒ s (four.7) Mutual implication: To show P ≡ Q, show P ⇒ Q and Q ⇒ P. (four.7.1) Reality implication: To show P, show true ⇒ P. (four.9) Proof by contradiction: To show P, show ¬P ⇒ f alse. (four.9.1) Proof by contradiction: To show P, show ¬P ≡ f alse. (four.12) Proof by contrapositive: To show P ⇒ Q, show ¬Q ⇒ ¬P.
Basic Legal guidelines of Quantification For symmetric and associative binary operator ⋆ with identification u. (eight.13) Axiom, Empty vary: (⋆x f alse : P) = u (eight.14) Axiom, One-point rule: Offered ¬happens(‘x’,‘E’),
(⋆x x = E : P) = P[x := E] (eight.15) Axiom, Distributivity: Offered P,Q : B or R is finite,
(⋆x R : P)⋆(⋆x R : Q) = (⋆x R : P ⋆ Q) (eight.16) Axiom, Vary break up: Offered R ∧ S ≡ f alse and P : B or R and S are finite,
(⋆x R ∨ S : P) = (⋆x R : P)⋆(⋆x S : P) (eight.17) Axiom, Vary break up: Offered P : B or R and S are finite,
(⋆x R ∨ S : P)⋆(⋆x R ∧ S : P) = (⋆x R : P)⋆(⋆x S : P) (eight.18) Axiom, Vary break up for idempotent ⋆ : Offered P : B or R and S are finite,
(⋆x R ∨ S : P) = (⋆x R : P)⋆(⋆x S : P) (eight.19) Axiom, Interchange of dummies: Offered ⋆ is idempotent or R and Q are finite,
¬happens(‘y’,‘R’), ¬happens(‘x’,‘Q’), (⋆x R : (⋆y Q : P)) = (⋆y Q : (⋆x R : P))
(eight.20) Axiom, nesting: Offered ¬happens(‘y’,‘R’), (⋆x,y R ∧ Q : P) = (⋆x R : (⋆y Q : P))
(eight.21) Axiom, Dummy renaming: Offered ¬happens(‘y’,‘R,P’), (⋆x R : P) = (⋆y R[x := y] : P[x := y])
(eight.22) Change of dummy: Offered ¬happens(‘y’,‘R,P’), and f has an inverse, (⋆x R : P) = (⋆y R[x := f .y] : P[x := f .y])
6 J. STANLEY WARFORD
(eight.23) Cut up off time period: For n: N, (a) (⋆i zero ≤ i < n + 1 : P) = (⋆i zero ≤ i < n : P)⋆ P[i := n] (b) (⋆i zero ≤ i < n + 1 : P) = P[i := 0]⋆(⋆i zero < i < n + 1 : P)
Theorems of the Predicate Calculus Common quantification. Notation: (⋆x : P) means (⋆x true : P). (9.2) Axiom, Buying and selling: (∀x R : P) ≡ (∀x : R ⇒ P) (9.three) Buying and selling:
(a) (∀x R : P) ≡ (∀x : ¬R ∨ P) (b) (∀x R : P) ≡ (∀x : R ∧ P ≡ R) (c) (∀x R : P) ≡ (∀x : R ∨ P ≡ P)
(9.four) Buying and selling: (a) (∀x Q ∧ R : P) ≡ (∀x Q : R ⇒ P) (b) (∀x Q ∧ R : P) ≡ (∀x Q : ¬R ∨ P) (c) (∀x Q ∧ R : P) ≡ (∀x Q : R ∧ P ≡ R) (d) (∀x Q ∧ R : P) ≡ (∀x Q : R ∨ P ≡ P)
(9.four.1) Common double buying and selling: (∀x R : P) ≡ (∀x ¬P : ¬R) (9.5) Axiom, Distributivity of ∨ over ∀ : Offered ¬happens(‘x’,‘P’),
P ∨(∀x R : Q) ≡ (∀x R : P ∨ Q) (9.6) Offered ¬happens(‘x’,‘P’), (∀x R : P) ≡ P ∨(∀x : ¬R) (9.7) Distributivity of ∧ over ∀ : Offered ¬happens(‘x’,‘P’),
¬(∀x : ¬R) ⇒ ((∀x R : P ∧ Q) ≡ P ∧(∀x R : Q)) (9.eight) (∀x R : true) ≡ true (9.9) (∀x R : P ≡ Q) ⇒ ((∀x R : P) ≡ (∀x R : Q)) (9.10) Vary weakening/strengthening: (∀x Q ∨ R : P) ⇒ (∀x Q : P) (9.11) Physique weakening/strengthening: (∀x R : P ∧ Q) ⇒ (∀x R : P) (9.12) Monotonicity of ∀ : (∀x R : Q ⇒ P) ⇒ ((∀x R : Q) ⇒ (∀x R : P)) (9.13) Instantiation: (∀x : P) ⇒ P[x := E] (9.16) Metatheorem: P is a theorem iff (∀x : P) is a theorem.
Existential quantification. (9.17) Axiom, Generalized De Morgan: (∃x R : P) ≡ ¬(∀x R : ¬P) (9.18) Generalized De Morgan:
(a) ¬(∃x R : ¬P) ≡ (∀x R : P) (b) ¬(∃x R : P) ≡ (∀x R : ¬P) (c) (∃x R : ¬P) ≡ ¬(∀x R : P)
(9.19) Buying and selling: (∃x R : P) ≡ (∃x : R ∧ P) (9.20) Buying and selling: (∃x Q ∧ R : P) ≡ (∃x Q : R ∧ P) (9.20.1) Existential double buying and selling: (∃x R : P) ≡ (∃x P : R)
THEOREMS FROM LADM 7
(9.20.2) (∃x : R) ⇒ ((∀x R : P) ⇒ (∃x R : P)) (9.21) Distributivity of ∧ over ∃ : Offered ¬happens(‘x’,‘P’),
P ∧(∃x R : Q) ≡ (∃x R : P ∧ Q) (9.22) Offered ¬happens(‘x’,‘P’), (∃x R : P) ≡ P ∧(∃x : R) (9.23) Distributivity of ∨ over ∃ : Offered ¬happens(‘x’,‘P’),
(∃x : R) ⇒ ((∃x R : P ∨ Q) ≡ P ∨(∃x R : Q)) (9.24) (∃x R : f alse) ≡ f alse (9.25) Vary weakening/strengthening: (∃x R : P) ⇒ (∃x Q ∨ R : P) (9.26) Physique weakening/strengthening: (∃x R : P) ⇒ (∃x R : P ∨ Q) (9.26.1) Physique weakening/strengthening: (∃x R : P ∧ Q) ⇒ (∃x R : P) (9.27) Monotonicity of ∃ : (∀x R : Q ⇒ P) ⇒ ((∃x R : Q) ⇒ (∃x R : P)) (9.28) ∃-Introduction: P[x := E] ⇒ (∃x : P) (9.29) Interchange of quantification: Offered ¬happens(‘y’,‘R’) and ¬happens(‘x’,‘Q’),
(∃x R : (∀y Q : P)) ⇒ (∀y Q : (∃x R : P)) (9.30) Offered ¬happens(‘x̂’,‘Q’),
(∃x R : P) ⇒ Q is a theorem iff (R ∧ P)[x := x̂] ⇒ Q is a theorem.
A Idea of Units (11.2) = x x = e0 ∨···∨ x = en−1 : x (11.three) Axiom, Set membership: Offered ¬happens(‘x’,‘F’),
F ∈ x R : E ≡ (∃x R : F = E) (11.four) Axiom, Extensionality: S = T ≡ (∀x : x ∈ S ≡ x ∈ T ) (11.four.1) Axiom, Empty set: /zero = x f alse : E (11.four.2) e ∈ /zero ≡ f alse (11.four.three) Axiom, Universe: U = x : x, U: set(t) = x: t : x (11.four.four) e ∈ U ≡ true, for e: t and U: set(t) (11.5) S = x x ∈ S : x (11.5.1) Axiom, Abbreviation: For x a single variable, = x R : x (11.6) Offered ¬happens(‘y’,‘R’) and ¬happens(‘y’,‘E’),
x R : E = (11.7) x ∈ ≡ R
R is the attribute predicate of the set. (11.7.1) y ∈ ≡ R[x := y] for any expression y (11.9) = ≡ (∀x : Q ≡ R) (11.10) = is legitimate iff Q ≡ R is legitimate. (11.11) Strategies for proving set equality S = T :
(a) Use Leibniz straight. (b) Use axiom Extensionality (11.four) and show the (9.eight) Lemma
v ∈ S ≡ v ∈ T for an arbitrary worth v. (c) Show Q ≡ R and conclude = .
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Operations on units. (11.12) Axiom, Dimension: #S = (Σx x ∈ S : 1) (11.13) Axiom, Subset: S ⊆ T ≡ (∀x x ∈ S : x ∈ T ) (11.14) Axiom, Correct subset: S ⊂ T ≡ S ⊆ T ∧ S ̸= T (11.15) Axiom, Superset: T ⊇ S ≡ S ⊆ T (11.16) Axiom, Correct superset: T ⊃ S ≡ S ⊂ T (11.17) Axiom, Complement: v ∈∼ S ≡ v ∈ U ∧ v /∈ S (11.18) v ∈∼ S ≡ v /∈ S, for v in U (11.19) ∼∼ S = S (11.20) Axiom, Union: v ∈ S ∪ T ≡ v ∈ S ∨ v ∈ T (11.21) Axiom, Intersection: v ∈ S ∩ T ≡ v ∈ S ∧ v ∈ T (11.22) Axiom, Distinction: v ∈ S − T ≡ v ∈ S ∧ v /∈ T (11.23) Axiom, Energy set: v ∈ PS ≡ v ⊆ S (11.24) Definition. Let Es be a set expression constructed from set variables, /zero, U, ∼, ∪, and ∩.
Then Ep is the expression constructed from Es by changing: /zero with f alse, U with true, ∪ with ∨, ∩ with ∧, ∼ with ¬. The building is reversible: Es will be constructed from Ep.
(11.25) Metatheorem. For any set expressions Es and Fs: (a) Es = Fs is legitimate iff Ep ≡ Fp is legitimate, (b) Es ⊆ Fs is legitimate iff Ep ⇒ Fp is legitimate, (c) Es = U is legitimate iff Ep is legitimate.
Primary properties of ∪. (11.26) Symmetry of ∪ : S ∪ T = T ∪ S (11.27) Associativity of ∪ : (S ∪ T )∪U = S ∪(T ∪U) (11.28) Idempotency of ∪ : S ∪ S = S (11.29) Zero of ∪ : S ∪ U = U (11.30) Id of ∪ : S ∪ /zero = S (11.31) Weakening: S ⊆ S ∪ T (11.32) Excluded center: S∪ ∼ S = U
Primary properties of ∩. (11.33) Symmetry of ∩ : S ∩ T = T ∩ S (11.34) Associativity of ∩ : (S ∩ T )∩U = S ∩(T ∩U) (11.35) Idempotency of ∩ : S ∩ S = S (11.36) Zero of ∩ : S ∩ /zero = /zero (11.37) Id of ∩ : S ∩ U = S (11.38) Strengthening: S ∩ T ⊆ S (11.39) Contradiction: S ∩ ∼ S = /zero
THEOREMS FROM LADM 9
Primary properties of combos of ∪ and ∩. (11.40) Distributivity of ∪ over ∩ : S ∪(T ∩U) = (S ∪ T )∩(S ∪U) (11.41) Distributivity of ∩ over ∪ : S ∩(T ∪U) = (S ∩ T )∪(S ∩U) (11.42) De Morgan:
(a) ∼ (S ∪ T ) = ∼ S ∩ ∼ T (b) ∼ (S ∩ T ) = ∼ S ∪ ∼ T
Extra properties of ∪ and ∩. (11.43) S ⊆ T ∧U ⊆ V ⇒ (S ∪U) ⊆ (T ∪V ) (11.44) S ⊆ T ∧U ⊆ V ⇒ (S ∩U) ⊆ (T ∩V ) (11.45) S ⊆ T ≡ S ∪ T = T (11.46) S ⊆ T ≡ S ∩ T = S (11.47) S ∪ T = U ≡ (∀x x ∈ U : x /∈ S ⇒ x ∈ T ) (11.48) S ∩ T = /zero ≡ (∀x : x ∈ S ⇒ x /∈ T )
Properties of set distinction. (11.49) S − T = S ∩ ∼ T (11.50) S − T ⊆ S (11.51) S − /zero = S (11.52) S ∩(T − S) = /zero (11.53) S ∪(T − S) = S ∪ T (11.54) S −(T ∪U) = (S − T )∩(S −U) (11.55) S −(T ∩U) = (S − T )∪(S −U)
Implication versus subset. (11.56) (∀x : P ⇒ Q) ≡ ⊆
Properties of subset. (11.57) Antisymmetry: S ⊆ T ∧ T ⊆ S ≡ S = T (11.58) Reflexivity: S ⊆ S (11.59) Transitivity: S ⊆ T ∧ T ⊆ U ⇒ S ⊆ U (11.60) /zero ⊆ S (11.61) S ⊂ T ≡ S ⊆ T ∧¬(T ⊆ S) (11.62) S ⊂ T ≡ S ⊆ T ∧(∃x x ∈ T : x /∈ S) (11.63) S ⊆ T ≡ S ⊂ T ∨ S = T (11.64) S ̸⊂ S (11.65) S ⊂ T ⇒ S ⊆ T (11.66) S ⊂ T ⇒ T ⊈ S (11.67) S ⊆ T ⇒ T ̸⊂ S (11.68) S ⊆ T ∧¬(U ⊆ T ) ⇒ ¬(U ⊆ S)
10 J. STANLEY WARFORD
(11.69) (∃x x ∈ S : x /∈ T ) ⇒ S ̸= T (11.70) Transitivity:
(a) S ⊆ T ∧ T ⊂ U ⇒ S ⊂ U (b) S ⊂ T ∧ T ⊆ U ⇒ S ⊂ U (c) S ⊂ T ∧ T ⊂ U ⇒ S ⊂ U
Theorems regarding energy set P. (11.71) P /zero = (11.72) S ∈ PS (11.73) #(PS) = 2#S (for finite set S)
Union and intersection of households of units. (11.74.1) Definition: v ∈ (∪x R : E) ≡ (∃x R : v ∈ E) (11.75.1) Definition: v ∈ (∩x R : E) ≡ (∀x R : v ∈ E) (11.76) Axiom, Partition: Set S partitions T if
(i) the units in S are pairwise disjoint and (ii) the union of the units in S is T , that’s, if (∀u,v u ∈ S ∧ v ∈ S ∧ u ̸= v : u ∩ v = /zero)∧(∪u u ∈ S : u) = T
Baggage. (11.80) Axiom, Dimension: # x R : E = (Σx R : 1) (11.81) Axiom, Variety of occurrences: v# x R : E = (Σx R ∧ v = E : 1) (11.82) Axiom, Bag equality: B = C ≡ (∀v : v#B = v#C) (11.83) Axiom, Subbag: B ⊆ C ≡ (∀v : v#B ≤ v#C) (11.84) Axiom, Correct subbag: B ⊂ C ≡ B ⊆ C ∧ B ̸= C (11.85) Axiom, Union: B ∪C = (11.86) Axiom, Intersection: B ∩C = v,i zero ≤ i < v#B ↓ v#C : v (11.87) Axiom, Distinction: B −C =
Mathematical Induction (12.three) Axiom, Mathematical Induction over N:
(∀n: N : (∀i zero ≤ i < n : P.i) ⇒ P.n) ⇒ (∀n: N : P.n) (12.four) Mathematical Induction over N:
(∀n: N : (∀i zero ≤ i < n : P.i) ⇒ P.n) ≡ (∀n: N : P.n) (12.5) Mathematical Induction over N:
P.zero ∧(∀n: N : (∀i zero ≤ i ≤ n : P.i) ⇒ P(n + 1)) ≡ (∀n: N : P.n) (12.11) Definition, b to the facility n:
b0 = 1 bn+1 = b · bn for n ≥ zero
THEOREMS FROM LADM 11
(12.12) b to the facility n: b0 = 1 bn = b · bn−1 for n ≥ 1
(12.13) Definition, factorial: zero! = 1 n! = n ·(n − 1)! for n > zero
(12.14) Definition, Fibonacci: F0 = zero, F1 = 1 Fn = Fn−1 + Fn−2 for n > 1
(12.14.1) Definition, Golden Ratio: ϕ = (1 + √
5)/2 ≈ 1.618 ϕ̂ = (1 − √
5)/2 ≈ −zero.618 (12.15) ϕ 2 = ϕ + 1 and ϕ̂ 2 = ϕ̂ + 1 (12.16) Fn ≤ ϕ n−1 for n ≥ 1 (12.16.1) ϕ n−2 ≤ Fn for n ≥ 1 (12.17) Fn+m = Fm · Fn+1 + Fm−1 · Fn for n ≥ zero and m ≥ 1
Inductively outlined binary timber. (12.30) Definition, Binary Tree:
/zero is a binary tree, referred to as the empty tree. (d,l,r) is a binary tree, for d: Z and l, r binary timber.
(12.31) Definition, Variety of Nodes: # /zero = zero #(d,l,r) = 1 + #l + #r
(12.32) Definition, Top: top./zero = zero top.(d,l,r) = 1 + max(top.l,top.r)
(12.32.1) Definition, Leaf: A leaf is a node with no kids (i.e. two empty subtrees). (12.32.2) Definition, Inside node: An inside node is a node that’s not a leaf. (12.32.three) Definition, Full: A binary tree is full if each node has both
zero or 2 kids. (12.33) The most variety of nodes in a tree with top n is 2n − 1 for n ≥ zero. (12.34) The minimal variety of nodes in a tree with top n is n for n ≥ zero. (12.35) (a) The most variety of leaves in a tree with top n is 2n−1 for n > zero.
(b) The most variety of inside nodes in a tree with top n is 2n−1 − 1 for n > zero. (12.36) (a) The minimal variety of leaves in a tree with top n is 1 for n > zero.
(b) The minimal variety of inside nodes in a tree with top n is n − 1 for n > zero. (12.37) Each nonempy full tree has an odd variety of nodes.
A Idea of Applications (p.1) Axiom, Excluded miracle: wp.S. false ≡ false (p.2) Axiom, Conjunctivity: wp.S.(X ∧Y ) ≡ wp.S.X ∧ wp.S.Y
12 J. STANLEY WARFORD
(p.three) Monotonicity: (X ⇒ Y ) ⇒ (wp.S.X ⇒ wp.S.Y ) (p.four) Definition, Hoare triple: S ≡ Q ⇒ wp.S.R (p.four.1) wp.S.R S (p.5) Postcondition rule: S ∧(A ⇒ R) ⇒ S (p.6) Definition, Program equivalence: S = T ≡ (For all R,wp.S.R ≡ wp.T.R) (p.7) (Q ⇒ A)∧ S ⇒ S (p.eight) Q0 S ∧ S ⇒ Q0 ∧ Q1 S R0 ∧ R1 (p.9) Q0 S ∧ S ⇒ S (p.10) Definition, skip: wp.skip.R ≡ R (p.11) skip ≡ Q ⇒ R (p.12) Definition, abort: wp.abort.R ≡ false (p.13) abort ≡ Q ≡ false (p.14) Definition, Composition: wp.(S; T ).R ≡ wp.S.(wp.T.R) (p.15) S ∧ T ⇒ S; T (p.16) Id of composition:
(a) S ; skip = S (b) skip ; S = S (p.17) Zero of composition:
(a) S ; abort = abort (b) abort ; S = abort (p.18) Definition, Task: wp.(x := E).R ≡ R[x := E] (p.19) Proof technique for task: (p.19) is (10.2)
To present that x := E is an implementation of x :=?, show Q ⇒ R[x := E].
(p.20) (x := x) = skip (p.21) IF G : (p.21) is (10.6)
if B1 → S1 [] B2 → S2 [] B3 → S3 fi
(p.22) Definition, IF G: wp.IF G.R ≡ (B1 ∨ B2 ∨ B3) ∧ B1 ⇒ wp.S1.R ∧ B2 ⇒ wp.S2.R ∧ B3 ⇒ wp.S3.R
(p.23) Empty guard: if fi = abort (p.24) Proof technique for IF G: (p.24) is (10.7)
To show IF G, it suffices to show (a) Q ⇒ B1 ∨ B2 ∨ B3, (b) S1 , (c) Q ∧ B2 S2 , and (d) S3 .
(p.25) ¬(B1 ∨ B2 ∨ B3) ⇒ IF G = abort (p.26) One-guard rule: if B → S fi ⇒ S (p.27) Distributivity of program over alternation:
if B1 → S1; T [] B2 → S2; T fi = if B1 → S1 [] B2 → S2 fi ; T
THEOREMS FROM LADM 13
(p.28) DO : do B → S od (p.29) Elementary Invariance Theorem. (p.29) is (12.43)
Suppose • P ∧ B S P holds—i.e. execution of S begun in a state
through which P and B are true terminates with P true—and • P do B → S od —i.e. execution of the loop begun
in a state through which P is true terminates. Then P do B → S od P ∧¬B holds.
(p.30) Proof technique for DO: (p.30) is (12.45) To show initialization; P do B → S od , it suffices to show (a) P is true earlier than execution of the loop: initialization; P, (b) P is a loop invariant: P ∧ B S P, (c) Execution of the loop terminates, and (d) R holds upon termination: P ∧¬B ⇒ R.
(p.31) False guard: do f alse → S od = skip
Relations and Capabilities (14.2) Axiom, Pair equality: ⟨b,c⟩ = ⟨b′,c′⟩ ≡ b = b′ ∧ c = c′ (14.2.1) Ordered pair one-point rule: Offered ¬happens(‘x,y’,‘E,F’),
(⋆x,y ⟨x,y⟩ = ⟨E,F⟩ : P) = P[x,y := E,F] (14.three) Axiom, Cross product: S × T = b,c b ∈ S ∧ c ∈ T : ⟨b,c⟩ (14.three.1) Axiom, Ordered pair extensionality:
U = V ≡ (∀x,y : ⟨x,y⟩ ∈ U ≡ ⟨x,y⟩ ∈ V )
Theorems for cross product. (14.four) Membership: ⟨x,y⟩ ∈ S × T ≡ x ∈ S ∧ y ∈ T (14.5) ⟨x,y⟩ ∈ S × T ≡ ⟨y,x⟩ ∈ T × S (14.6) S = /zero ⇒ S × T = T × S = /zero (14.7) S × T = T × S ≡ S = /zero ∨ T = /zero ∨ S = T (14.eight) Distributivity of × over ∪ :
(a) S ×(T ∪U) = (S × T )∪(S ×U) (b) (S ∪ T )×U = (S ×U)∪(T ×U)
(14.9) Distributivity of × over ∩ : (a) S ×(T ∩U) = (S × T )∩(S ×U) (b) (S ∩ T )×U = (S ×U)∩(T ×U)
(14.10) Distributivity of × over − : S ×(T −U) = (S × T )−(S ×U)
(14.11) Monotonicity: T ⊆ U ⇒ S × T ⊆ S ×U (14.12) S ⊆ U ∧ T ⊆ V ⇒ S × T ⊆ U ×V
14 J. STANLEY WARFORD
(14.13) S × T ⊆ S ×U ∧ S ̸= /zero ⇒ T ⊆ U (14.14) (S ∩ T )×(U ∩V ) = (S ×U)∩(T ×V ) (14.15) For finite S and T , #(S × T ) = #S · #T
Relations. (14.15.1) Definition, Binary relation:
A binary relation over B ×C is a subset of B ×C. (14.15.2) Definition, Id: The identification relation iB on B is iB = (14.15.three) Id lemma: ⟨x,y⟩ ∈ iB ≡ x = y (14.15.four) Notation: ⟨b,c⟩ ∈ ρ and b ρ c are interchangeable notations. (14.15.5) Conjunctive which means: b ρ c σ d ≡ b ρ c ∧ c σ d The area Dom.ρ and vary Ran.ρ of a relation ρ on B ×C are outlined by (14.16) Definition, Area: Dom.ρ = (14.17) Definition, Vary: Ran.ρ = The inverse ρ−1 of a relation ρ on B ×C is the relation outlined by (14.18) Definition, Inverse: ⟨b,c⟩ ∈ ρ−1 ≡ ⟨c,b⟩ ∈ ρ, for all b: B, c: C (14.19) Let ρ and σ be relations.
(a) Dom(ρ−1) = Ran.ρ (b) Ran(ρ−1) = Dom.ρ (c) If ρ is a relation on B ×C, then ρ−1 is a relation on C × B (d) (ρ−1)−1 = ρ (e) ρ ⊆ σ ≡ ρ−1 ⊆ σ −1
Let ρ be a relation on B ×C and σ be a relation on C × D. The product of ρ and σ, denoted by ρ ◦ σ, is the relation outlined by (14.20) Definition, Product: ⟨b,d⟩ ∈ ρ ◦ σ ≡ (∃c c ∈ C : ⟨b,c⟩ ∈ ρ ∧⟨c,d⟩ ∈ σ) or, utilizing the choice notation by (14.21) Definition, Product: b (ρ ◦ σ) d ≡ (∃c : b ρ c σ d)
Theorems for relation product. (14.22) Associativity of ◦ : ρ ◦(σ ◦ θ ) = (ρ ◦ σ)◦ θ (14.23) Distributivity of ◦ over ∪ :
(a) ρ ◦(σ ∪ θ ) = (ρ ◦ σ)∪(ρ ◦ θ ) (b) (σ ∪ θ )◦ ρ = (σ ◦ ρ)∪(θ ◦ ρ)
(14.24) Distributivity of ◦ over ∩ : (a) ρ ◦(σ ∩ θ ) ⊆ (ρ ◦ σ)∩(ρ ◦ θ ) (b) (σ ∩ θ )◦ ρ ⊆ (σ ◦ ρ)∩(θ ◦ ρ)
THEOREMS FROM LADM 15
Theorems for powers of a relation. (14.25) Definition:
ρ zero = iB ρ n+1 = ρ n ◦ ρ for n ≥ zero
(14.26) ρ m ◦ ρ n = ρ m+n for m ≥ zero,n ≥ zero (14.27) (ρ m)n = ρ m·n for m ≥ zero,n ≥ zero (14.28) For ρ a relation on finite set B of n components,
(∃i, j zero ≤ i < j ≤ 2n 2
: ρ i = ρ j) (14.29) Let ρ be a relation on a finite set B. Suppose ρ i = ρ j and zero ≤ i < j. Then
(a) ρ i+ok = ρ j+ok for ok ≥ zero (b) ρ i = ρ i+p·( j−i) for p ≥ zero
Desk 14.1 Lessons of relations ρ over set B Identify Property Different
(a) reflexive (∀b : b ρ b) iB ⊆ ρ (b) irreflexive (∀b : ¬(b ρ b)) iB ∩ ρ = /zero (c) symmetric (∀b,c : b ρ c ≡ c ρ b) ρ−1 = ρ (d) antisymmetric (∀b,c : b ρ c ∧ c ρ b ⇒ b = c) ρ ∩ ρ−1 ⊆ iB (e) uneven (∀b,c : b ρ c ⇒ ¬(c ρ b)) ρ ∩ ρ−1 = /zero (f) transitive (∀b,c,d : b ρ c ∧ c ρ d ⇒ b ρ d) ρ = (∪i i > zero : ρ i)
(14.30.1) Definition: Let ρ be a relation on a set. The reflexive closure of ρ is the relation r(ρ) that satisfies: (a) r(ρ) is reflexive; (b) ρ ⊆ r(ρ); (c) If any relation σ is reflexive and ρ ⊆ σ, then r(ρ) ⊆ σ.
(14.30.2) Definition: Let ρ be a relation on a set. The symmetric closure of ρ is the relation s(ρ) that satisfies: (a) s(ρ) is symmetric; (b) ρ ⊆ s(ρ); (c) If any relation σ is symmetric and ρ ⊆ σ, then s(ρ) ⊆ σ.
(14.30.three) Definition: Let ρ be a relation on a set. The transitive closure of ρ is the relation ρ+ that satisfies: (a) ρ+ is transitive; (b) ρ ⊆ ρ+; (c) If any relation σ is transitive and ρ ⊆ σ, then ρ+ ⊆ σ.
(14.30.four) Definition: Let ρ be a relation on a set. The reflexive transitive closure of ρ is the relation ρ∗ that’s each the reflexive and the transitive closure of ρ.
16 J. STANLEY WARFORD
(14.31) (a) A reflexive relation is its personal reflexive closure. (b) A symmetric relation is its personal symmetric closure. (c) A transitive relation is its personal transitive closure.
(14.32) Let ρ be a relation on a set B. Then, (a) r(ρ) = ρ ∪ iB (b) s(ρ) = ρ ∪ ρ−1 (c) ρ+ = (∪i zero < i : ρ i) (d) ρ∗ = ρ+ ∪ iB
Equivalence relations. (14.33) Definition: A relation is an equivalence relation iff it’s reflexive, symmetric,
and transitive (14.34) Definition: Let ρ be an equivalence relation on B. Then [b]ρ, the equivalence
class of b, is the subset of components of B which might be equal (beneath ρ) to b: x ∈ [b]ρ ≡ x ρ b
(14.35) Let ρ be an equivalence relation on B, and let b, c be members of B. The following three predicates are equal: (a) b ρ c (b) [b]∩[c] ̸= /zero (c) [b] = [c] That’s, (b ρ c) = ([b]∩[c] ̸= /zero) = ([b] = [c])
(14.35.1) Let ρ be an equivalence relation on B. The equivalence lessons partition B. (14.36) Let P be the set of units of a partition of B. The following relation ρ on B is an
equivalence relation: b ρ c ≡ (∃p p ∈ P : b ∈ p ∧ c ∈ p)
Capabilities. (14.37) (a) Definition: A binary relation f on B ×C is determinate iff
(∀b,c,c′ b f c ∧ b f c′ : c = c′) (b) Definition: A binary relation is a operate iff it’s determinate.
(14.37.1) Notation: f .b = c and b f c are interchangeable notations. (14.38) Definition: A operate f on B ×C is complete if B = Dom. f .
In any other case it’s partial. We write f : B → C for the kind of f if f is complete and f : B ; C if f is partial.
(14.38.1) Whole: A operate f on B ×C is complete if, for an arbitrary aspect b: B, (∃c: C : f .b = c)
(14.39) Definition, Composition: For capabilities f and g, f • g = g ◦ f . (14.40) Let g : B → C and f : C → D be complete capabilities.
Then the composition f • g of f and g is the entire operate outlined by ( f • g).b = f (g.b)
THEOREMS FROM LADM 17
ρ a relation on B ×C f a operate, f : B → C
Determinate (14.37) Whole (14.38)
B C B C
Determinate: f is a operate Whole
B C B C
Not determinate: ρ isn’t a operate Not complete (partial)
Onto (14.41a) One-to-one (14.41b)
B C B C
Onto One-to-one
B C B C
Not onto Not one-to-one
Inverses of complete capabilities. (14.41) Definitions:
(a) Whole operate f : B → C is onto or surjective if Ran. f = C. (b) Whole operate f is one-to-one or injective if
(∀b,b′: B,c: C : b f c ∧ b′ f c ≡ b = b′). (c) Whole operate f is bijective whether it is one-to-one and onto.
(14.42) Let f be a complete operate, and let f −1 be its relational inverse. (a) Then f −1 is a operate, i.e. is determinate, iff f is one-to-one. (b) And, f −1 is complete iff f is onto.
(14.43) Definitions: Let f : B → C. (a) A left inverse of f is a operate g : C → B such that g • f = iB. (b) A proper inverse of f is a operate g : C → B such that f • g = iC. (c) Perform g is an inverse of f whether it is each a left inverse and a proper inverse.
(14.44) Perform f : B → C is onto iff f has a proper inverse. (14.45) Let f : B → C be complete. Then f is one-to-one iff f has a left inverse.
18 J. STANLEY WARFORD
(14.46) Let f : B → C be complete. The following statements are equal. (a) f is one-to-one and onto. (b) There’s a operate g : C → B that’s each a left and a proper inverse of f . (c) f has a left inverse and f has a proper inverse.
Order relations. (14.47) Definition: A binary relation ρ on a set B is named a partial order on b whether it is
reflexive, antisymmetric, and transitive. In this case, pair ⟨B,ρ⟩ is named a partially ordered set or poset.
We use the image ⪯ for an arbitrary partial order, typically writing c ⪰ b as an alternative of b ⪯ c. (14.47.1) Definition, Incomparable: incomp(b,c) ≡ ¬(b ⪯ c)∧¬(c ⪯ b) (14.48) Definition: Relation ≺ is a quasi order or strict partial order if ≺ is transitive
and irreflexive (14.48.1) Definition, Reflexive discount: Given ⪯, its reflexive discount ≺ is computed
by eliminating all pairs ⟨b,b⟩ from ⪯. (14.48.2) Let ≺ be the reflexive discount of ⪯. Then,
¬(b ⪯ c) ≡ c ≺ b ∨ incomp(b,c) (14.49) (a) If ρ is a partial order over a set B, then ρ − iB is a quasi order.
(b) If ρ is a quasi order over a set B, then ρ ∪ iB is a partial order.
Whole orders and topological kind. (14.50) Definition: A partial order ⪯ over B is named a complete or linear order if
(∀b,c : b ⪯ c ∨ b ⪰ c), i.e. iff ⪯ ∪ ⪯−1= B × B. In this case, the pair ⟨B,⪯⟩ is named a linearly ordered set or a sequence.
(14.51) Definitions: Let S be a nonempty subset of poset ⟨U,⪯⟩. (a) Aspect b of S is a minimal aspect of S if no aspect of S is smaller than b,
i.e. if b ∈ S ∧(∀c c ≺ b : c /∈ S). (b) Aspect b of S is the least aspect of S if b ∈ S ∧(∀c c ∈ S : b ⪯ c). (c) Aspect b is a decrease certain of S if (∀c c ∈ S : b ⪯ c).
(A decrease certain of S needn’t be in S.) (d) Aspect b is the best decrease certain of S, written glb.S if b is a decrease certain
and if each decrease certain c satisfies c ⪯ b. (14.52) Each finite nonempty subset S of poset ⟨U,⪯⟩ has a minimal aspect. (14.53) Let B be a nonempty subset of poset ⟨U,⪯⟩.
(a) A least aspect of B can be a minimal aspect of B (however not essentially vice versa).
(b) A least aspect of B can be a best decrease certain of B (however not essentially vice versa).
THEOREMS FROM LADM 19
(c) A decrease certain of B that belongs to B can be a least aspect of B.
((14.54) Definitions: Let S be a nonempty subset of poset ⟨U,⪯⟩. (a) Aspect b of S is a maximal aspect of S if no aspect of S is bigger than b,
i.e. if b ∈ S ∧(∀c b ≺ c : c /∈ S). (b) Aspect b of S is the best aspect of S if b ∈ S ∧(∀c c ∈ S : c ⪯ b). (c) Aspect b is an higher certain of S if (∀c c ∈ S : c ⪯ b).
(An higher certain of S needn’t be in S.) (d) Aspect b is the least higher certain of S, written lub.S, if b is an higher certain
and if each higher certain c satisfies b ⪯ c.
Relational databases. (14.56.1) Definition, choose: For Relation R and predicate F, which can include names
of fields of R, σ(R,F) = t t ∈ R ∧ F (14.56.2) Definition, venture: For A1,…,Am a subset of the names of the fields of
relation R, π(R,A1,…,Am) = t t ∈ R : ⟨t.A1,t.A2,…,t.Am⟩ (14.56.three) Definition, pure be a part of: For Relations R1 and R2, R1 ▷◁ R2 has all of the attributes
that R1 and R2 have, but when an attribute seems in each, then it seems solely as soon as within the end result; additional, solely these tuples that agree on this widespread attribute are included.
Development of Capabilities (g.1) Definition of asymptotic higher certain: For a given operate g.n, O(g.n),
pronounced “big-oh of g of n”, is the set of capabilities f .n (∃c,n0 c > zero ∧ n0 > zero : (∀n n ≥ n0 : zero ≤ f .n ≤ c · g.n) )
(g.2) O-notation: f .n = O(g.n) means operate f .n is within the set O(g.n). (g.three) Definition of asymptotic decrease certain: For a given operate g.n, Ω (g.n),
pronounced “big-omega of g of n”, is the set of capabilities
(g.four) Ω-notation: f .n = Ω (g.n) means operate f .n is within the set Ω (g.n). (g.5) Definition of asymptotic tight certain: For a given operate g.n, Θ (g.n),
pronounced “big-theta of g of n”, is the set of capabilities f .n (∃c1,c2,n0 c1 > zero ∧ c2 > zero ∧ n0 > zero :
(∀n n ≥ n0 : zero ≤ c1 · g.n ≤ f .n ≤ c2 · g.n) ) (g.6) Θ-notation: f .n = Θ (g.n) means operate f .n is within the set Θ (g.n). (g.7) f .n = Θ (g.n) if and provided that f .n = O(g.n) and f .n = Ω (g.n)
20 J. STANLEY WARFORD
Comparability of capabilities. (g.eight) Reflexivity:
(a) f .n = O( f .n) (b) f .n = Ω ( f .n) (c) f .n = Θ ( f .n)
(g.9) Symmetry: f .n = Θ (g.n) ≡ g.n = Θ ( f .n) (g.10) Transpose symmetry: f .n = O(g.n) ≡ g.n = Ω ( f .n) (g.11) Transitivity:
(a) f .n = O(g.n) ∧ g.n = O(h.n) ⇒ f .n = O(h.n) (b) f .n = Ω (g.n) ∧ g.n = Ω (h.n) ⇒ f .n = Ω (h.n) (c) f .n = Θ (g.n) ∧ g.n = Θ (h.n) ⇒ f .n = Θ (h.n)
(g.12) Outline an asymptotically constructive polynomial p.n of diploma d to be p.n = (Σi zero ≤ i ≤ d : aini) the place the constants a0,a1,…,advert are the coefficients of the polynomial and advert > zero. Then p.n = Θ (nd).
(g.13) (a) O(1) ⊂ O(lg n) ⊂ O(n) ⊂ O(n lg n) ⊂ O(n2) ⊂ O(n3) ⊂ O(2n) (b) Ω (1) ⊃ Ω (lg n) ⊃ Ω (n) ⊃ Ω (n lg n) ⊃ Ω (n2) ⊃ Ω (n3) ⊃ Ω (2n)
A Idea of Integers Minimal and most. (15.53) Definition of ↓ : (∀z : z ≤ x ↓ y ≡ z ≤ x ∧ z ≤ y)
Definition of ↑ : (∀z : z ≥ x ↑ y ≡ z ≥ x ∧ z ≥ y) (15.54) Symmetry:
(a) x ↓ y = y ↓ x (b) x ↑ y = y ↑ x (15.55) Associativity:
(a) (x ↓ y) ↓ z = x ↓ (y ↓ z) (b) (x ↑ y) ↑ z = x ↑ (y ↑ z)
Restrictions. Though ↓ and ↑ are symmetric and associative, they don’t have identities over the integers. Due to this fact, axiom (eight.13) empty vary doesn’t apply to ↓ or ↑. Additionally, when utilizing range-split axioms, no vary ought to be false.
(15.56) Idempotency: (a) x ↓ x = x (b) x ↑ x = x
Divisibility. (15.77) Definition of | : c | b ≡ (∃d : c · d = b) (15.78) c | c (15.79) c | zero (15.80) 1 | b (15.80.1) −b | c ≡ b | c (15.80.2) −1 | b
THEOREMS FROM LADM 21
(15.81) c | 1 ⇒ c = 1 ∨ c = −1 (15.81.1) c | 1 ≡ c = 1 ∨ c = −1 (15.82) d | c ∧ c | b ⇒ d | b (15.83) b | c ∧ c | b ≡ b = c ∨ b = −c (15.84) b | c ⇒ b | c · d (15.85) b | c ⇒ b · d | c · d (15.86) 1 < b ∧ b | c ⇒ ¬(b | (c + 1)) (15.87) Theorem: Given integers b, c with c > zero, there exist (distinctive) integers q and r
such that b = q · c + r, the place zero ≤ r < c. (15.89) Corollary: For given b, c, the values q and r of Theorem (15.87) are distinctive.
Best widespread divisor. (15.90) Definition of ÷ and mod for operands b and c, c ̸= zero :
b ÷ c = q, b mod c = r the place b = q · c + r and zero ≤ r < c (15.91) b = c ·(b ÷ c)+ b mod c for c ̸= zero (15.92) Definition of gcd:
b gcd c = (↑ d d | b ∧ d | c : d) for b, c not each zero zero gcd zero = zero
(15.94) Definition of lcm : b lcm c = (↓ ok: Z+ b | ok ∧ c | ok : ok) for b ̸= zero and c ̸= zero b lcm c = zero for b = zero or c = zero
Properties of gcd. (15.96) Symmetry: b gcd c = c gcd b (15.97) Associativity: (b gcd c) gcd d = b gcd (c gcd d) (15.98) Idempotency: (b gcd b) = abs.b (15.99) Zero: 1 gcd b = 1 (15.100) Id: zero gcd b = abs.b (15.101) b gcd c = (abs.b) gcd (abs.c) (15.102) b gcd c = b gcd (b + c) = b gcd (b − c) (15.103) b = a · c + d ⇒ b gcd c = c gcd d (15.104) Distributivity: d ·(b gcd c) = (d · b) gcd (d · c) for zero ≤ d (15.105) Definition of comparatively prime ⊥ : b ⊥ c ≡ b gcd c = 1 (15.107) Inductive definition of gcd:
b gcd zero = b b gcd c = c gcd (b mod c)
(15.108) (∃x,y : x · b + y · c = b gcd c) for all b,c: N (15.111) ok | b ∧ ok | c ≡ ok | (b gcd c)
22 J. STANLEY WARFORD
Combinatorial Assessment (16.1) Rule of sum: The measurement of the union of n (finite) pairwise disjoint units is the
sum of their sizes. (16.2) Rule of product: The measurement of the cross product of n units is the product of
their sizes. (16.three) Rule of distinction: The measurement of a set with a subset of it eliminated is the scale of
the set minus the scale of the subset. (16.four) Definition: P(n,r) = n!/(n − r)! (16.5) The variety of r-permutations of a set of measurement n equals P(n,r). (16.6) The variety of r-permutations with repetition of a set of measurement n is nr. (16.7) The variety of permutations of a bag of measurement n with ok distinct components occurring
n1,n2,…,nk instances is n!
n1! · n2! ····· nk! .
(16.9) Definition: The binomial coefficient (n
r
) , which is learn as “n select r”, is
outlined by (
n r
) =
n! r! ·(n − r)!
for zero ≤ r ≤ n.
(16.10) The variety of r-combinations of n components is (n
r
) .
(16.11) The quantity (n
r
) of r-combinations of a set of measurement n equals the variety of
permutations of a bag that comprises r copies of 1 object and n − r copies of one other.
A Idea of Graphs (19.1) Definition: Let V be a finite, nonempty set and E a binary relation on V .
Then G = ⟨V,E⟩ is named a directed graph, or digraph. A component of V is named a vertex; a component of E is named an edge.
(19.1.1) Definitions: (a) In an undirected graph ⟨V,E⟩, E is a set of unordered pairs. (b) In a multigraph ⟨V,E⟩, E is a bag of undirected edges. (c) The indegree of a vertex of a digraph is the variety of edges for which it’s
an finish vertex. (d) The outdegree of a vertex of a digraph is the variety of edges for which it’s
a begin vertex. (e) The diploma of a vertex is the sum of its indegree and outdegree. (f) An edge ⟨b,b⟩ for some vertex b is a self-loop. (g) A digraph with no self-loops is named loop-free.
(19.three) The sum of the levels of the vertices of a digraph or multigraph equals 2 · #E. (19.four) In a digraph or multigraph, the variety of vertices of wierd diploma is even.
THEOREMS FROM LADM 23
(19.four.1) Definition: A path has the next properties. (a) A path begins with a vertex, ends with a vertex, and alternates between
vertices and edges. (b) Every directed edge in a path is preceded by its begin vertex and adopted by
its finish vertex. An undirected edge is preceded by one in all its vertices and adopted by the opposite.
(c) No edge seems greater than as soon as. (19.four.2) Definitions:
(a) A easy path is a path through which no vertex seems greater than as soon as, besides that the primary and final vertices often is the identical.
(b) A cycle is a path with at the very least one edge, and with the primary and final vertices the identical.
(c) An undirected multigraph is linked if there’s a path between any two vertices.
(d) A digraph is linked if making its edges undirected ends in a linked multigraph.
(19.6) If a graph has a path from vertex b to vertex c, then it has a easy path from b to c. (19.6.1) Definitions:
(a) An Euler path of a multigraph is a path that comprises every fringe of the graph precisely as soon as.
(b) An Euler circuit is an Euler path whose first and final vertices are the identical. (19.eight) An undirected linked multigraph has an Euler circuit iff each vertex has even
diploma. (19.eight.1) Definitions:
(a) A full graph with n vertices, denoted by Kn, is an undirected, loop- free graph in which there’s an edge between each pair of distinct vertices.
(b) A bipartite graph is an undirected graph through which the set of vertices are partitioned into two units X and Y such that every edge is incident on one vertex in X and one vertex in Y .
(19.10) A path of a bipartite graph is of even size iff its ends are in the identical partition aspect.
(19.11) A linked graph is bipartite iff each cycle has even size. (19.11.1) Definition: A full bipartite graph Km,n is a bipartite graph through which
one partition aspect X has m vertices, the opposite partition aspect Y has n vertices, and there’s an edge between every vertex of X and every vertex of Y .
(19.11.2) Definitions: (a) A Hamilton path of a graph or digraph is a path that comprises every vertex
precisely as soon as, besides that the top vertices of the trail often is the identical. (b) A Hamilton circuit is a Hamilton path that could be a cycle.
24 J. STANLEY WARFORD
Pure Science Division, Pepperdine College, Malibu, CA 90263 Electronic mail deal with: Stan.Warford@pepperdine.edu URL: https://www.cslab.pepperdine.edu/warford/
