Let R and S be two partial orders on the same set X. Considering R and S as subsets of X ×X, we

Let R and S be two partial orders on the same set X....

Key Concepts

Partial Order

A partial order is a binary relation on a set that is reflexive, antisymmetric, and transitive. It provides a way to compare elements in a set where not every pair of elements may be comparable. This concept is central to order theory and many applications in mathematics and computer science.

Reflexivity

Reflexivity is a property of a relation where every element is related to itself. In the context of partial orders, this ensures that for any element x in the set, the ordered pair (x, x) is included in the relation.

Antisymmetry

Antisymmetry is the condition that if one element x is related to an element y and y is also related to x, then x and y must be the same element. This property avoids the possibility of having two distinct elements that are mutually comparable, which is crucial for maintaining the structure of a partial order.

Transitivity

Transitivity in a relation means that if an element x is related to y and y is related to z, then x must necessarily be related to z. For partial orders, transitivity ensures that the ordering is consistent and that indirect relationships are covered.

Extension of Relations

The extension of a relation involves adding new ordered pairs to an existing relation while attempting to preserve the defining properties (in this case, reflexivity, antisymmetry, and transitivity). When extending a partial order, one must be careful to add pairs that do not violate these properties, especially transitivity and antisymmetry.

Set Inclusion

Set inclusion in the context of binary relations refers to one relation being a subset of another. This concept helps in comparing two orders on the same set, as seen when one partial order is contained within another but is not equal. It is a useful tool for analyzing how one can extend a partial order by adding new pairs from a larger relation.

Transcript

00:01 In this question, we are asked to show that lecical graphic order is a partial ordering on the set of strings from pose set.

00:11 So here it doesn't mean just characters or number, it's any postset, yes, with its own partial ordering.

00:23 We will have element in s, right? suppose call it a1, a2, and so on.

00:30 String here mean any combination of elements that we want.

00:39 Suppose we write this in some combination.

00:44 And this will be a string of elements from opposite.

00:51 And the set of all such words are what we are considering.

00:57 So let's go over the cycle graphic.

01:01 For a string a and b, i will call it a.

01:06 1 to a and b1 to b m and let the way we compare two words is that first if they are equal so they are the same then we are done we have a and b related the other way is that it's strictly one thing is strictly less than the other in this case we will look at the minimum of their length and one of the two things have to happen for a to be less than b either within within the minimal length a is strictly less than b or they are equal but the word b itself is longer than a so the second case it may be that suppose for english letter we have a b, c, c, d, and a, b, and that's it.

02:19 If we have it like this, then the second word is less than the first word just because the link is lower.

02:31 Okay, the two position here are the same.

02:38 And that's how we have, let me delete this.

02:43 That's how we have relations.

02:45 Either they are equal or one of the two things happens.

02:49 So let's go over the property of partial ordering.

02:54 First, reflexive, obviously, any words will relate to itself.

03:00 Antisymmetry, suppose two words a and b have their relation both ways.

03:07 So i will use contradicting argument that if they are not, not equal when these two things happen but they are not equal then we must have either one or two which are both impossible one is that the string of the the string of length minimal like of length key are both less than and greater than each other which is impossible right the other is that between the minimal length the way words are the same, but then the length of a is both lower and greater than the length of b, which is not possible as well.

04:02 So by supposing this, we are led to impossible situation, so it's not true from the start.

04:13 It must, we must have that a equal to b...

Please give Ace some feedback