Let ⟨A, R⟩be a well-ordering, and B be a nonempty subset of A. Define RB on B : for any x, y, RB

Let ⟨A, R⟩be a well-ordering, and B be a nonempty subset of...

Key Concepts

Well-Ordering

A well-ordering is an ordered set in which every nonempty subset has a least element. This concept is central to theories involving transfinite induction and recursion, ensuring that no infinite descending sequence exists and that each subset contains a minimal element.

Restriction of an Order Relation

The restriction of an order relation to a subset involves defining the order on the subset by inheriting order relations from the larger set. This concept is important when examining whether order-theoretic properties, such as well-ordering, are maintained when focusing on a particular subset.

Transcript

00:01 Hello there.

00:02 So let's consider that we got a set s and this set has an elements, okay? and there are elements, so let's say ab are part of this set s and we should check how many relations are exist on s such that in the first instance ab this set this pair is on this relation okay so how to do that well we need to to consider pairs of numbers so we're going to consider here for example as one as two elements on s times s so here we are going to have an square elements you can use the product rule to obtain this value because you got n possibilities for the first position and allowing to repetitions that means that in the second the second element of this pair can also have n possibilities so this means that we obtain n to square possible values for these pairs okay great so we got n square possibilities for having this kind of pairs, s1, s2 on s cross s, where s1 and s2 are part of this set s.

02:07 So we got an possible pairs.

02:12 So how many of them are different from this set, ab? okay so that translates to taking n square minus one that is this element in particular and we left with n minus one order pairs that that are different from ab and from these pairs we got two possibilities either this this element is part of r or not okay so that means is that each one has two choices.

02:56 So each of these elements here has two possibilities.

03:00 So we are going to multiply this 2 to 2 2 2 n -square minus 1 times, but that translates to take 2 to the n -square minus 1.

03:19 So how many of ab are on how many of these kind of relations? let's remember that a is different from b is one of the assumptions.

03:34 So the number is 2 to the n squared minus 1 in this case.

03:42 Then for the part v, we should check those pairs, ab, that are not elements of are not relations.

03:53 So the procedure is really the same.

03:56 As i mentioned before, we got n square minus 1 element different.

04:02 From this pair ab.

04:06 And we got two possibilities for these elements.

04:10 They can be part of r or not be part of this relation.

04:18 So it's the same procedure, we obtain the same value actually.

04:22 Here we multiply 2 to 2 to 2, exactly n squared minus 1 times and this is equal to 2 to the n square minus 1 so this is the solution for the second for the part v then we should check those pairs that uh not so how many relations are exist on s such that no other pair has a okay, so how to check this? well, we got, as i mentioned, we got in principle n square pairs, order pairs, okay? and so we should count how many of them...

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