# 4.2: Existence of SDRs - Mathematics

In this section, sdr means complete sdr. It is easy to see that not every collection of sets has an sdr. Consider

$$A_1={a,b}, A_2={a,b}, A_3={a,b}, A_4={b,c,d,e}.$$

Now the total number of possible representatives is 5, and we only need 4. Nevertheless, this is impossible, because the first three sets have no sdr considered by themselves. Thus the following condition, called Hall's Condition, is clearly necessary for the existence of an sdr: For every (kge1), and every set ({i_1,i_2,ldots,i_k}subseteq [n]), (|igcup_{j=1}^k A_{i_j}|ge k). That is, the number of possible representatives in any collection of sets must be at least as large as the number of sets. Both examples fail to have this property because (|A_1cup A_2cup A_3|=2< 3).

Remarkably, this condition is both necessary and sufficient.

Hall's Theorem

A collection of sets (A_1,A_2,ldots,A_n) has an sdr if and only if for every (kge1), and every set ({i_1,i_2,ldots,i_k}subseteq [n]), (|igcup_{j=1}^k A_{i_j}|ge k).

Proof

We already know the condition is necessary, so we prove sufficiency by induction on (n).

Suppose (n=1); the condition is simply that (|A_1|ge 1). If this is true then (A_1) is non-empty and so there is an sdr. This establishes the base case.

Now suppose that the theorem is true for a collection of (k< n) sets, and suppose we have sets (A_1,A_2,ldots,A_n) satisfying Hall's Condition. We need to show there is an sdr.

Suppose first that for every (k< n) and every ({i_1,i_2,ldots,i_k}subseteq [n]), that (|igcup_{j=1}^k A_{i_j}|ge k+1), that is, that these unions are larger than required. Pick any element (x_nin A_n), and define (B_i=A_iackslash{x_n}) for each (i< n). Consider the collection of sets (B_1,ldots,B_{n-1}), and any union (igcup_{j=1}^k B_{i_j}) of a subcollection of the sets. There are two possibilities: either (igcup_{j=1}^k B_{i_j}=igcup_{j=1}^k A_{i_j}) or (igcup_{j=1}^k B_{i_j}=igcup_{j=1}^k A_{i_j}ackslash{x_n}), so that (|igcup_{j=1}^k B_{i_j}|=|igcup_{j=1}^k A_{i_j}|) or (|igcup_{j=1}^k B_{i_j}|=|igcup_{j=1}^k A_{i_j}|-1). In either case, since (|igcup_{j=1}^k A_{i_j}|ge k+1), (|igcup_{j=1}^k B_{i_j}|ge k). Thus, by the induction hypothesis, the collection (B_1,ldots,B_{n-1}) has an sdr ({x_1,x_2,ldots,x_{n-1}}), and for every (i< n), (x_i ot= x_n), by the definition of the (B_i). Thus ({x_1,x_2,ldots,x_{n}}) is an sdr for (A_1,A_2,ldots,A_n).

If it is not true that for every (k< n) and every ({i_1,i_2,ldots,i_k}subseteq [n]), (|igcup_{j=1}^k A_{i_j}|ge k+1), then for some (k< n) and ({i_1,i_2,ldots,i_k}), (|igcup_{j=1}^k A_{i_j}|= k). Without loss of generality, we may assume that (|igcup_{j=1}^k A_{j}|= k). By the induction hypothesis, (A_1,A_2,ldots,A_k) has an sdr, ({x_1,ldots,x_k}).

Define (B_i=A_iackslash igcup_{j=1}^k A_{j}) for (i> k). Suppose that ({x_{k+1},ldots,x_{n}}) is an sdr for (B_{k+1},ldots,B_{n}); then it is also an sdr for (A_{k+1},ldots,A_{n}). Moreover, ({x_1,ldots,x_n}) is an sdr for (A_{1},ldots,A_{n}). Thus, to finish the proof it suffices to show that (B_{k+1},ldots,B_{n}) has an sdr. The number of sets here is (n-k< n), so we need only show that the sets satisfy Hall's Condition.

So consider some sets (B_{i_1},B_{i_2},…,B_{i_l}). First we notice that $$|A_1cup A_2cupcdotscup A_kcup B_{i_1}cup B_{i_2}cupcdots B_{i_l}|=k+|B_{i_1}cup B_{i_2}cupcdots B_{i_l}|.$$ Also $$|A_1cup A_2cupcdotscup A_kcup B_{i_1}cup B_{i_2}cupcdots B_{i_l}|=|A_1cup A_2cupcdotscup A_kcup A_{i_1}cup A_{i_2}cupcdots A_{i_l}|$$ and $$|A_1cup A_2cupcdotscup A_kcup A_{i_1}cup A_{i_2}cupcdots A_{i_l}|ge k+l.$$ Putting these together gives eqalign{ k+|B_{i_1}cup B_{i_2}cupcdotscup B_{i_l}|&ge k+lcr |B_{i_1}cup B_{i_2}cupcdotscup B_{i_l}|&ge lcr. } Thus, (B_{k+1},ldots,B_{n}) has an sdr, which finishes the proof.

(square)

## 4.2: Existence of SDRs - Mathematics

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1. What is Combinatorics?

1.1 Example: Perfect Covers of Chessboards

1.3 Example: The Four-Color Problem

1.4 Example: The Problem of the 36 Officers

1.5 Example: Shortest-Route Problem

1.6 Example: Mutually Overlapping Circles

1.7 Example: The Game of Nim

2. The Pigeonhole Principle

2.1 Pigeonhole Principle: Simple Form

2.2 Pigeonhole Principle: Strong Form

3. Permutations and Combinations

3.1 Four Basic Counting Principles

3.4 Permutations of Multisets

3.5 Combinations of Multisets

4. Generating Permutations and Combinations

4.1 Generating Permutations

4.2 Inversions in Permutations

4.3 Generating Combinations

4.4 Generating r-Combinations

4.5 Partial Orders and Equivalence Relations

5. The Binomial Coefficients

5.3 Unimodality of Binomial Coefficients

5.4 The Multinomial Theorem

5.5 Newton's Binomial Theorem

5.6 More on Partially Ordered Sets

6. The Inclusion-Exclusion Principle and Applications

6.1 The Inclusion-Exclusion Principle

6.2 Combinations with Repetition

6.4 Permutations with Forbidden Positions

6.5 Another Forbidden Position Problem

7. Recurrence Relations and Generating Functions

7.3 Exponential Generating Functions

7.4 Solving Linear Homogeneous Recurrence Relations

7.5 Nonhomogeneous Recurrence Relations

8. Special Counting Sequences

8.2 Difference Sequences and Stirling Numbers

8.5 Lattice Paths and Schröder Numbers

9. Systems of Distinct Representatives

9.1 General Problem Formulation

10. Combinatorial Designs

10.3 Steiner Triple Systems

11. Introduction to Graph Theory

11.3 Hamilton Paths and Cycles

11.4 Bipartite Multigraphs

11.6 The Shannon Switching Game

12. More on Graph Theory

12.2 Plane and Planar Graphs

12.4 Independence Number and Clique Number

13. Digraphs and Networks

13.3 Matching in Bipartite Graphs Revisited

14.1 Permutation and Symmetry Groups

14.3 Pólya's Counting formula

## 3. Objections to Mathematical Platonism

A variety of objections to mathematical platonism have been developed. Here are the most important ones.

### 3.1 Epistemological access

The most influential objection is probably the one inspired by Benacerraf (1973). What follows is an improved version of Benacerraf&rsquos objection due to Field (1989).  This version relies on the following three premises.

 Premise 1. Mathematicians are reliable, in the sense that for almost every mathematical sentence S, if mathematicians accept S, then S is true. Premise 2. For belief in mathematics to be justified, it must at least in principle be possible to explain the reliability described in Premise 1. Premise 3. If mathematical platonism is true, then this reliability cannot be explained even in principle.

If these three premises are correct, it will follow that mathematical platonism undercuts our justification for believing in mathematics.

But are the premises correct? The first two premises are relatively uncontroversial. Most platonists are already committed to Premise 1. And Premise 2 seems fairly secure. If the reliability of some belief formation procedure could not even in principle be explained, then the procedure would seem to work purely by chance, thus undercutting any justification we have for the beliefs produced in this way.

Premise 3 is far more controversial. Field defends this premise by observing that &ldquothe truth-values of our mathematical assertions depend on facts involving platonic entities that reside in a realm outside of space-time&rdquo (Field 1989, p. 68) and thus are causally isolated from us even in principle. However, this defense assumes that any adequate explanation of the reliability in question must involve some causal correlation. This has been challenged by a variety of philosophers who have proposed more minimal explanations of the reliability claim. (See Burgess & Rosen 1997, pp. 41&ndash49 and Lewis 1991, pp. 111&ndash112 cf. also Clarke-Doane 2016. See Linnebo 2006 for a critique.) 

### 3.2 A metaphysical objection

Another famous article by Benacerraf develops a metaphysical objection to mathematical platonism (Benacerraf 1965, cf. also Kitcher 1978). Although Benacerraf focuses on arithmetic, the objection naturally generalizes to most pure mathematical objects.

Benacerraf opens by defending what is now known as a structuralist view of the natural numbers, according to which the natural numbers have no properties other than those they have in virtue of being positions in an &omega-sequence. For instance, there is nothing more to being the number 3 than having certain intrastructurally defined relational properties, such as succeeding 2, being half of 6, and being prime. No matter how hard we study arithmetic and set theory, we will never know whether 3 is identical with the fourth von Neumann ordinal, or with the corresponding Zermelo ordinal, or perhaps, as Frege suggested, with the class of all three-membered classes (in some system that allows such classes to exist).

Benacerraf now draws the following conclusion:

In other words, Benacerraf claims that there can be no objects which have nothing but structural properties. All objects must have some non-structural properties as well. (See Benacerraf 1996 for some later reflections on this argument.)

Both of the steps of Benacerraf&rsquos argument are controversial. The first step&mdashthat natural numbers have only structural properties&mdashhas recently been defended by a variety of mathematical structuralists (Parsons 1990, Resnik 1997, and Shapiro 1997). But this step is denied by logicists and neo-logicists, who claim that the natural numbers are intrinsically tied to the cardinalities of the collections that they number. And the second step&mdashthat there can be no objects with only structural properties&mdashis explicitly rejected by all of the structuralists who defend the first step. (For some voices sympathetic to the second step, see Hellman 2001 and MacBride 2005. See also Linnebo 2008 for discussion.)

### 3.3 Other metaphysical objections

In addition to Benacerraf&rsquos, a variety of metaphysical objections to mathematical platonism have been developed. One of the more famous examples is an argument of Nelson Goodman&rsquos against set theory. Goodman (1956) defends the Principle of Nominalism, which states that whenever two entities have the same basic constituents, they are identical. This principle can be regarded as a strengthening of the familiar set theoretic axiom of extensionality. The axiom of extensionality states that if two sets x and y have the same elements&mdashthat is, if &forallu(u &isin x &harr u &isin y)&mdashthen they are identical. The Principle of Nominalism is obtained by replacing the membership relation with its transitive closure.  The principle thus states that if x and y are borne &isin* by the same individuals&mdashthat is, if &forallu(u &isin* x &harr u &isin* y)&mdashthen x and y are identical. By endorsing this principle, Goodman disallows the formation of sets and classes, allowing only the formation of mereological sums and the application to the standard mereological operations (as described by his &ldquocalculus of individuals&rdquo).

However, Goodman&rsquos defense of the Principle of Nominalism is now widely held to be unconvincing, as witnessed by the widespread acceptance by philosophers and mathematicians of set theory as a legitimate and valuable branch of mathematics.

## Simple math shows how many space aliens may be out there If you say you believe space aliens exist, I doubt your friends will be shocked. In a universe aglow with 2 trillion galaxies, you'd be supremely smug to think that Earth alone hosts clever creatures. One 2015 poll showed that 54 percent of Americans feel confident that intelligent aliens are out there.

Maybe that optimism comes from science fiction. After all, if there are no extraterrestrials, there isn't much of a mission for the Starship Enterprise or job openings for Vulcans. Fiction aside, many scientists agree that the cosmos is undoubtedly sprinkled — perhaps liberally sprinkled — with life. Even sentient life.

But can we say anything about that sprinkling? Can we hazard a guess as to how close the nearest aliens might be?

This is an uncertain business but not a new one. In 1961 astronomer Frank Drake devised a simple equation for estimating the number of "technically active" societies in our galaxy. That bit of easy math is known as the Drake Equation, and it's often said to be the second most famous formula in science (the first being Einstein's E = mc2).

If you look up the formula online, you'll see that it takes into account the odds that there are habitable planets around other stars, the likelihood that life will arise, and the probability that biology will occasionally evolve to produce clever beings. But even without wrestling with the Drake Equation, we can use similar reasoning to gauge the plentitude of alien societies and how close the Klingons might be.

We start with recent research showing that one in six stars hosts a planet hospitable to life. No, not one in a million. One in six. So let's take that number and run with it. Next we have to make a few assumptions. In particular, if you were given a million Earth-size worlds, what fraction do you think would ever beget technically sophisticated inhabitants?

Life on our planet began quickly: random chemical activity in 350 million trillion gallons of ocean water spawned a reproducing molecule within a few hundred million years. So maybe biology doesn't need much of a goad to get started. I don't think it's unreasonable to figure that at least half of all planets suitable for life actually produce it.

Intelligence is less certain. The dinosaurs were a good design but didn't do well in school. But let's say that one in 100 biology-encrusted planets eventually coughs up some thinking beings. And, as per Frank Drake, let's also assume that any Klingons out there continue to hang out for 10,000 years before self-destructing (nuclear war, anyone?) or meeting some other woeful end.

### Mach We just beamed a signal at space aliens. Was that a bad idea?

Do the arithmetic, and you'll find that one in 100 million star systems has technically adept inhabitants. That's not much different than the fraction of jackpot tickets in this week's Powerball lottery.

So how close are the nearest signaling extraterrestrials? If we're going to pay good money to fire up the warp drive and visit some bumpy-headed aliens, how far do we have to travel? Well, the average distance between stars in our part of the galaxy is 4.2 light-years (the distance to Proxima Centauri). That is, for every cube of space that's 4.2 light-years on a side, you'll find (on average) one star. Now imagine a bigger box, 2,000 light-years on a side. It will contain 100 million star boxes, and one sophisticated civilization.

By this rough and ready calculation, the nearest aliens are probably between one and two thousand light-years away. In other words, no closer than the three bright stars of Orion's Belt. Sure, alien neighbors might be farther — or closer. But this order-of-magnitude estimate tells us that they're not next door. They haven't heard our news reports, and they're not likely to have any incentive to visit. They simply don't know we're here.

By the way, we probably aren't going to visit them either. Today's fastest rockets would take at least 20 million years to get there, by which time you're going to be awfully tired of on-board pretzels.

Yes, the aliens are likely around, and 10,000 societies could inhabit our galaxy (not to mention those other galaxies!) They're not close. But they may be discoverable. That's why we continue to search the sky for radio signals launched into the ether long ago by our cosmic brethren.

## 4.2: Existence of SDRs - Mathematics

All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited.

Feature Papers represent the most advanced research with significant potential for high impact in the field. Feature Papers are submitted upon individual invitation or recommendation by the scientific editors and undergo peer review prior to publication.

The Feature Paper can be either an original research article, a substantial novel research study that often involves several techniques or approaches, or a comprehensive review paper with concise and precise updates on the latest progress in the field that systematically reviews the most exciting advances in scientific literature. This type of paper provides an outlook on future directions of research or possible applications.

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to authors, or important in this field. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

## Role of Self-Efficacy Toward Students’ Achievement in Mathematical Multiple Representation Ability (MMRA)

Various studies showed that there are some points which affect students’ achievement one of them is mathematics self-efficacy. Self-efficacy is feeling, trust, and confidence of affective behavior of students toward their ability. In learning process, the students find obstacles in solving their problem such as relating one representation to another representation. Previous studies indicated that students’ achievement of mathematical representation ability have obstacles because the students did not understand relationship between concepts, ideas, and materials to be represented. The other studies claimed that the higher students’ self-efficacy, the higher mathematical multiple representation ability (MMRA). It means that self-efficacy has positive correlation with mathematic ability. This article will discuss about role of mathematics self-efficacy toward ability achievement of mathematical multiple representation ability (MMRA) in solving mathematics problem in order to increase students’ achievement.

## Square Root of 2

Let's look at the square root of 2 more closely.

 When we draw a square of size "1", what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950. (etc)

But it is not a number like 3, or five-thirds, or anything like that .

. in fact we cannot write the square root of 2 using a ratio of two numbers

. I explain why on the Is It Irrational? page,

. and so we know it is an irrational number

The simplest example is probably the polynomial $X^8+1$, and now I’ll explain why.

It’s a fact that every finite abelian extension of $mathbb Q$ is contained in a cyclotomic extension $mathbb Q(zeta_m)$, $zeta_m$ being a primitive $m$-th root of unity. The Galois group of $mathbb Q(zeta_m)$ over $mathbb Q$ is the group of units of the ring $mathbb Z/mmathbb Z$. With this information, you can get any finite abelian group as the quotient of a $(mathbb Z/mmathbb Z)^ imes$, and so an extension of $mathbb Q$ with that Galois group.

Now, $mathbb Z/16mathbb Z$ has for its units the odd numbers modulo $16$, as a multiplicative group, and you easily check that this has the shape $mathbb Z/2mathbb Z imesmathbb Z/4mathbb Z$, generators being $-1$ and $5$. The polynomial for the primitive sixteenth roots of unity is $(X^<16>-1)/(X^8-1)=X^8+1$, and there you have it.

Consider the splitting field of $x^5-1$, $x^2-2$ on $mathbb$. A) The Galois group of a degree $n$ polynomial need not be a transitive subgroup of $S_n$. E.g. what is the Galois group of a biquadratic extension? What you have in mind is that the Galois group of an irreducible polynomial is a transitive subgroup of $S_n$, where $n$ is the degree of the polynomial.

B) Why does a Galois group of order 8 have to be a subgroup of $S_4$? What tells you that you cannot have, say, a cyclic group of order 8 as a Galois group (which would then be a transitive subgroup of $S_8$)?

So with these remarks in mind, perhaps you should ask the precise question that you meant to ask.

Anticipating what your question might be: MAGMA tells me that the group generated by $(1, 4)(2, 5)(3, 8)(6, 7), (1, 5, 6, 8)(2, 7, 3, 4), ext (1, 6)(2, 3)(4, 7)(5, 8)$ is a transitive subgroup of $S_8$ that is isomorphic to $C_2 imes C_4$. It also tells me that there are no such subgroups in any $S_n$ for $n$ smaller than 8.