Last Update: 30 November 2010
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a_{n} = 3a_{n–1} – 2a_{n–2}, ∀ n ≥ 2
∴ α_{1} = –α_{2}
∴ 2(–α_{2}) + α_{2} = 1
∴ –α_{2} = 1
∴ α_{2} = –1
∴ α_{1} = 1
set-theoretical notation: | (a,b) ∈ R |
FOL notation: | "R(a,b)" for: (a,b) ∈ R "¬R(a,b)" for: (a,b) ∉ R |
mathematical notation: | "aRb" for (a,b) ∈ R |
Then:
e.g.) (Buffalo, NY) ∈ R
i.e.) R(Buffalo, NY)
We could also have a 3-place ("ternary") relation Z ⊆
C×S×ZIP s.t. (Buffalo, NY, 14260) ∈ Z
or: Z(Buffalo, NY, 14260)
or, recursively (where S(y) = y+1):
Base Case: | (0,1) ∈ < |
Recursive Case #1: | (x,y) ∈ < → (x,S(y)) ∈ < |
Recursive Case #2: | (x,y) ∈ < → (S(x),S(y)) ∈ < |
e.g.)
Let #isa be the URI for {(x,y) ∈ U×U | x is a y},
where U = universe (i.e., domain).
Let #Dolphin be the URI for the category of dolphins.
Then:
rel'n | ref? | sym? | anti-sym? | trans? | equiv? |
---|---|---|---|---|---|
≤ | √ | × | √ | √ | × |
≥ | √ | × | √ | √ | × |
< | × | × | √ vacuously! | √ | × |
> | × | × | √ vacuously! | √ | × |
= | √ | √ | √ | √ | √ |