Last Update: 27 April 2009
Note: or material is highlighted |
but probably not:
and certainly not:
and t_{i} are terms (noun phrases),
then P(t_{1}, …, t_{n}) is an atomic proposition
Universal Instantiation & Generalization
Existential Instantiation & Generalization
i.e., [ƒ(a) = b ∧ ƒ(a) = b′ → b = b′]
i.e., no two objects in range have the same pre-image
i.e., (∀a, a′ ∈ A)[ƒ(a) = ƒ(a′) → a = a′]
i.e., if 2 outputs are the same, then their inputs were the same
i.e., no two objects in domain have the same image
(∀ set S ⊆ N) [( (0 ∈ S) ∧ (∀k ∈ N) [k ∈ S → k+1 ∈ S] ) → S = N]
(∀ property P) [( P(0) ∧ (∀k ∈ N)[P(k) → P(k+1)]) → (∀n ∈ N)P(n) ]
To show (∀n ∈ N)P(n):
Let r be a vertex different from any of these r_{i}.
Then the graph consisting of r with an edge to each r_{i} is a rooted tree with root r.
r² – c_{1}r – c_{2} = 0
i. | R is reflexive | =def | (∀a ∈ A)R(a, a) |
ii. | R is symmetric | =def | (∀a, b ∈ A)[R(a, b) → R(b, a)] |
iii. | R is anti-symmetric | =def | (∀a, b ∈ A)[R(a, b) ∧ R(b, a) → a = b] |
iv. | R is transitive | =def | (∀a, b, c ∈ A)[R(a, b) ∧ R(b, c) → R(a, c)] |
R is reflexive | ↔ | every vertex has a loop |
R is symmetric | ↔ | for each edge, there is an inverse edge |
R is anti-symmetric | ↔ | the only pairs of inverse edges are loops. |
R is transitive | ↔ | every path of 2 edges has a shortcut. |
Let G be a connected, planar, simple graph
that
divides the plane into a set R of regions.
Then: