Sufficient and Necessary

-

 A is necessary for B means:

A must be true whenever B is true


e.g, being a mammal (clause A) is necessary to be a human (clause B)


in order words, if human, then mammal, 

or if not mammal, then not human

so if B then A




---



A is sufficient for B means:

whenever A is satisfied, B must be satisfied


e.g. if some number is rational (clause A), then it is sufficient for that number to be a real number (clause B)


or in order words, if A, then B




---



so if you want if A and B  AND   if B then A   to be true simultaneously


then A must be sufficient for B, and A must be necessary for B


e.g.

A: today is fourth of July

B: today is independence Day in USA


if today is fourth of July, it is sufficient to infer that today is independence day  ( A => B )


if today is fourth of July is necessary for today to be independence day

which is another way of saying, ( B => A )




---



so wait, you might ask


why are both sufficient and necessary conditions qualities of clause A?

why don't we define necessary in terms of some condition of clause B?


no idea, but that's just how it is. 



on the flip side, you can now express everything concisely. 


Instead of having to say, A is sufficient for B AND B is sufficient for A, thus A <=> B


you can now say A is necessary and sufficient for B, thus A <=>B


which means, A is both enough to show B, and is a consequence of B




---



NOTE: A is sufficient for B   is equivalent to   B is necessary for A


so if you say, A is sufficient for B, then by trivial logic, B is necessary for A, meaning B must happen whenever A happens. 




Comments

Post a Comment

Popular posts from this blog

How does Entropy work to split Decision Trees?

-
How to determine which split is best for a decision tree Entropy! The one with lowest entropy wins. Think of entropy as the "impurity" of a group. Less entropy = more pure. More entropy = more chaos = less pure.  Note: -log(p) = information. We want to maximize this while Entropy = - sum (p*log(p)).  Turns out, when you add the probability in, and sum ALL possibilities in a group, the lower the better. It's just how the numbers work out.  Btw, if group is homogenous, i.e. only one possible outcome, then entropy = 0, no randomness by def.       Note: could use Gini index instead, easier calculation, Gini = 1- p^2 for all p, in group, then weighted average of all outcomes from a split, same with Entropy)      (computationally faster, but quite a bit, that's why R uses it as the default, and gives almost  the same answer as Entropy, but empirically slightly worse splits, i.e. lower accuracy on testing datasets) (also rec...
Read more

Idiosyncrasies of Modulo Arithmetic

-
 Modulo Arithmetic  (two main ways, technically there's a weird 3rd but we're gonna ignore that) Mathematical Definition: Original, by Gauss' definition in his 1801 magnum opus, mod is defined as: given integers a, b a ≡ b (mod n) (read as a is congruent to b modulo n) means a-b is an integer multiple of n so for something like a = 13, b = 63, n = 10 a-b = -50 which is a multiple of 10, thus it works so a logical extension would be,  13 mod 10 = 3k, and 63 mod 10 = 3k so any multiple of 3 would be a valid solution to a mod n but since it's equivalent to a and b having the same "remainder" (quotes because remaining has two definitions, which will be come clear very soon) we have the following more useful modern definition: for some dividend N divisor D quotient Q remainder R N = D * Q + R where |R| < |D| Intuitively,  suppose a situation of N people, and D groups with Q people per group with R remaining then, number of groups (D) * people per group (Q) + rem...
Read more

Lumen Candela Lux Nits

-
 1. Understand Steradian You know how exactly 2PI of a circle's radius makes up the the circumference of a circle Fact: this also means the angle subtended at the center of the circle by an arc of length 1 radius is exactly 1 radian ~ 57.3 degrees Now, in the 3D case, the surface area of a sphere is 4 * PI * r^2 and if we use the same logic, and analogous to using r as our base unit in the 2D case, we use r^2, we can say that 1 steradian the angle subtended by the surface areas of exactly is exactly 1 r^2 meaning the entire sphere has solid angle of 4*PI steradians Quick Recap:  1 radius (r) subtends an angle of 1 radian 1 surface patch (r^2) subtends a solid angle of 1 steradian (solid angle = angle but 3D, steradian = radian but 3D, surface patch = radius but 2D) (yes, 2D) 2. Candela (cd) Just note, that 1 common candle emits roughly 1 candela (cd) of luminous intensity (that's the actual term) If you think of light as tangible, and flying in all directions then if you c...
Read more