mt 1 for discrete math

Axiom of Equality: Reflexivity

∀a : a = a

Axiom of Equality: Symmetry

∀a, b : a = b ⇒ b = a

Axiom of Equality: Transitivity

∀a, b, c : [a = b ∧ b = c] ⇒ a = c

Axiom of ℕ/ℤ: Non-triviality

0 ≠ 1

Axiom of ℕ/ℤ: Compatibility of Addition

∀a, b, c : a = b ⇒ a + c = b + c

Axiom of ℕ/ℤ: Compatibility of Multiplication

∀a, b, c : a = b ⇒ ac = bc

Axiom of ℕ/ℤ: Associativity of Addition

∀a, b, c : (a + b) + c = a + (b + c)

Axiom of ℕ/ℤ: Associativity of Multiplication

∀a, b, c : (ab)c = a(bc)

Axiom of ℕ/ℤ: Commutativity of Addition

∀a, b : a + b = b + a

Axiom of ℕ/ℤ: Commutativity of Multiplication

∀a, b : ab = ba

Axiom of ℕ/ℤ: Additive Identity

∀a : a + 0 = a

Axiom of ℕ/ℤ: Multiplicative Identity

∀a : a · 1 = a

Axiom of ℕ/ℤ: Distributivity

∀a, b, c : a(b + c) = ab + ac

Axiom of ℤ: Integrality (Zero Product)

∀a, b : ab = 0 ⇒ [a = 0 ∨ b = 0]

Axiom of ℤ: Additive Invertibility

∀a, ∃b : a + b = 0

Order Axiom: Reflexivity

∀a : a ≤ a

Order Axiom: Antisymmetry

∀a, b : [a ≤ b ∧ b ≤ a] ⇒ a = b

Order Axiom: Transitivity

∀a, b, c : [a ≤ b ∧ b ≤ c] ⇒ a ≤ c

Order Axiom: Totality

∀a, b : a ≤ b ∨ b ≤ a

Order Axiom: Compatibility of Addition

∀a, b, c : a ≤ b ⇒ a + c ≤ b + c

Order Axiom: Compatibility of Multiplication

∀a, b : [a ≥ 0 ∧ b ≥ 0] ⇒ ab ≥ 0

Logic Law: Idempotence

P ∧ P ≡ P and P ∨ P ≡ P

Logic Law: Domination

P ∧ F ≡ F and P ∨ T ≡ T

Logic Law: Identity

P ∧ T ≡ P and P ∨ F ≡ P

Logic Law: Double Negation

¬(¬P) ≡ P

Logic Law: DeMorgan’s Laws

¬(P ∧ Q) ≡ ¬P ∨ ¬Q and ¬(P ∨ Q) ≡ ¬P ∧ ¬Q

Definition of Implication

P ⇒ Q ≡ ¬P ∨ Q

Logic Law: Negation of Implication

¬(P ⇒ Q) ≡ P ∧ ¬Q

Logic Law: Contrapositive Equivalence

P ⇒ Q ≡ ¬Q ⇒ ¬P

Logic Law: Distributivity

P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)

Negation of Universal Quantifier

¬(∀x P(x)) ≡ ∃x ¬P(x)

Negation of Existential Quantifier

¬(∃x P(x)) ≡ ∀x ¬P(x)

Well-Ordering Principle

Every non-empty set of natural numbers has a least element