a) Let R be the relation on the set of functions from 𝐙^+ to 𝐙^+ such that (f, g) belongs to R i

A) Let R be the relation on the set of functions from 𝐙^+ to...

Key Concepts

Equivalence Relation

An equivalence relation on a set is a relation that satisfies three key properties: reflexivity, symmetry, and transitivity. This means that every element is related to itself, if one element is related to another then the reverse holds, and if one element is related to a second and the second to a third, then the first is related to the third. These properties allow the set to be partitioned into disjoint subsets known as equivalence classes, where each element of the set belongs to exactly one class where all elements are equivalent under the relation.

Theta Notation

Theta (?) notation is a symbol used in asymptotic analysis to describe the asymptotically tight bound on the growth rate of functions. It provides a way to formally express that two functions have the same order of growth, meaning that they grow at the same rate up to constant factors. This concept is essential in computer science and mathematics for comparing the efficiency of algorithms or the behavior of functions as their input goes to infinity.

Equivalence Class

An equivalence class under an equivalence relation is the subset of elements that are all equivalent to one another. In the context of functions and growth rates, an equivalence class consists of all functions that grow asymptotically at the same rate, as characterized by the Theta notation. Understanding equivalence classes helps in categorizing functions based on their behavior for large inputs, thereby simplifying the analysis of their growth patterns.

Transcript

00:01 In part a, we're asked to show that the relation on the set of functions from the positive managers to the positive managers, such that the pair f and g belongs to this relation, if and only f is big theta of g, is an equivalence relation.

00:24 So to prove this is an equivalence relation, let's prove it's reflexive.

00:32 So let f be a function from the positive managers to the positive managers.

00:50 And as f is large data, take k to be 1, then we have that for all x greater than k with c equal to 1, we have that c times the absolute value of f of x.

01:31 Well, this is simply going to be the absolute value of f of x.

01:34 So this is going to be less than or equal to the absolute value of f of x.

01:50 Clearly, this is going to be less than or equal to one times the absolute value of f of x, which is c times the absolute value of f of x.

02:00 So this implies that f is big theta of f, and therefore we have that relation to r is reflexive, since f is related to itself.

02:25 Now to prove the r is symmetric, we're going to, going to pose that pair fg is an r.

02:39 This means that f is big theta of g.

02:51 So then we have that for all x greater than k, for some non -zero constants, c1, c2, and k, we have that c1 times the absolute value of g of x, is less than or equal to absolute value of f of x, which is less than equal to c2 times the absolute value of g of x.

03:34 This is what being b theta of g means.

03:39 Now let's take c1 prime to be 1 over c1, and let's take c2 prime to be 1 over c2.

03:57 This is possible because c1 and c2 are both non -zero constants.

04:08 Then we have for all x greater than k by solving.

04:15 The above inequality, well, we have the absolute value of g of x over c1 prime is less than or equal to the absolute value of f of x, which is less than or equal to absolute value of g of x over c2 prime.

04:44 And so we have that, in other words, c1 prime, absolute type of x of x, is going to be less than or equal to absolute value of g of x, which is less than or equal to c2 prime times the absolute value of f of x.

05:35 Actually in reverse, we have, sorry, c2 prime times f of x.

05:40 This is going to be less than equal to g of x, which is less than or equal to c1 prime, f of x.

05:47 My mistake.

05:52 So by big theta notation, we have that g is big theta of f.

06:01 So if you're a big theta of f, follows that g is related to f and therefore that r is symmetric.

06:16 Now to prove that r is transitive, suppose that f and g are related and that g and h are related.

06:34 So this implies that f is big theta of g and we have that g is big theta of h...

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