sh -c 'exec strace -p $$'
strace: I'm sorry, I can't let you do that, Dave.
My name's not Dave!
An IPv4 designation of /n provides 232-n addresses, 0 ≤ n ≤ 32. A subnet mask of 255.255.255.0 is no different than "/24."
An IPv6 designation of /n provides 2128-n addresses, 0 ≤ n ≤ 128. For example, a /48 contains 2128-48 addresses, which is equal to 280 or about 1.2 x 1024.
In some contexts, "192.168.1.0/24" is a set (in the set theory sense) of all IP addresses in the 192.168.1.0 to 192.168.1.255 range. For example, 192.168.0.0/24 ∪ 192.168.1.0/24 = 192.168.0.0/23.
|IPv4 only||Dual Stack||IPv6 only|
|IPv4 only||connects via IPv4||connects via IPv4||cannot connect|
|Server||Dual Stack||connects via IPv4||connects via IPv4 or IPv6||connects via IPv6|
|IPv6 only||cannot connect||connects via IPv6||connects via IPv6|
Proof by induction
H(n) is some predicate of a natural number n.
If H(0) is true,
and the material conditional H(m) -> H(n), 0 <= m < n for all n > 0 is true
then H(n) is true for all n >= 0.
- H(0) & (H(0) -> H(1)) therefore H(1).
- H(1) & (H(1) -> H(2)) therefore H(2).
- H(2) & (H(2) -> H(3)) therefore H(3).
- and so on.
In other words, you do not prove H(n) is true for all n directly. Rather, you prove H(0) (the basis step) and then you prove the material conditional H(n) -> H(n+1) (inductive step), and by continuous modus ponens, you prove H(n) for all n in N (natural numbers).
For example, prove that every power of five greater than 5 ends in 25. That is, 5^n = 25 (mod 100) for all integer n >= 2.
Base case: n = 2. 5^2 = 25 (mod 100).
Inductive step: Given 5^n = 25 (mod 100), prove 5^(n+1) = 25 (mod 100).
- 5^(n+1) (mod 100)
- = 5^n * 5 (mod 100)
- = 25 * 5 (mod 100) by the inductive hypothesis
- = 125 (mod 100)
- = 25 (mod 100).
- People sometimes think that my website is full of nonsense. Not that people disagree with the views presented, but rather that it's full of my computer networking niche.
- In a similar vein, people often think that I'm childish because I sort of like doing these nonsensical things just to get people's attention.