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Posted on March 22, 2024 by expressvnp中文官网

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Figurate numbers and forward differences

There are lots of further directions this could be taken but I’ll leave that to you and your kids. I tried to create something that was conducive to open-ended exploration rather than something that had a single particular goal in mind.

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Posted on March 17, 2024 by Brent

You have a function $f : A \to B$ and want to prove it is a bijection. What can you do?

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A bijection is defined as a function which is both one-to-one and onto. So prove that $expressvnp中文官网$ is one-to-one, and prove that it is onto.

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If $A$ and $expressvnp中文官网$ are finite and have the same size, it’s enough to prove expressvnp中文官网 that $f$ is one-to-one, or that $f$ is onto. A one-to-one function between two finite sets of the same size must also be onto, and vice versa. (Of course, if $A$ and $expressvnp中文官网$ don’t have the same size, then there can’t possibly be a bijection between them in the first place.)

Intuitively, this makes sense: on the one hand, in order for $f$ to be onto, it “can’t afford” to send multiple elements of $A$ to the same element of $B$ , because then it won’t have enough to cover every element of $B$ . So it must be one-to-one. Likewise, in order to be one-to-one, it can’t afford to miss any elements of $B$ , because then the elements of $A$ have to “squeeze” into fewer elements of $B$ , and some of them are bound to end up mapping to the same element of $expressvnp中文官网$ . So it must be onto.

However, this is actually kind of tricky to formally prove! Note that the definition of “ $expressvnp中文官网$ and $expressvnp中文官网$ have the same size” is that there exists some bijection $expressvnp中文官网$ . A proof has to start with a one-to-one (or onto) function $f$ , and some expressvnp中文官网 bijection $g$ , and somehow prove that $f$ is onto (or one-to-one). Also, a valid proof must somehow account for the fact that this becomes false when $expressvnp中文官网$ and $B$ are infinite: a one-to-one function between two infinite sets of the same size need not be onto, or vice versa; we saw several examples in my previous post, such as $expressvnp中文官网$ defined by $expressvnp中文官网$ . Although tricky to come up with, the proof is cute and not too hard to understand once you see it; I think I may write about it in another post!

Note that we can even relax the condition on sizes a bit further: for example, it’s enough to prove that $expressvnp中文官网$ is one-to-one, and the finite size of $expressvnp中文官网$ is greater than or equal to the finite size of $expressvnp中文官网$ . The point is that $expressvnp中文官网$ being a one-to-one function implies that the size of $A$ is less than or equal to the size of $B$ , so in fact they have equal sizes.

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One can also prove that $f : A \to B$ is a bijection by showing that it has an inverse: a function $expressvnp中文官网$ such that $g(f(a)) = a$ and $f(g(b)) = b$ for all $a \in A$ and $expressvnp中文官网$ . As we saw in my last post, these facts imply that $f$ is one-to-one and onto, and hence a bijection. And it really is necessary to prove both $g(f(a)) = a$ and $f(g(b)) = b$ : if only one of these hold then $g$ is called a left or right inverse, respectively (more generally, a one-sided inverse), but $f$ needs to have a full-fledged two-sided inverse in order to be a bijection.

…unless $A$ and $B$ are of the same finite size! In that case, it’s enough to show the existence of a one-sided inverse—say, a function $g$ such that $g(f(a)) = a$ . Then $f$ is (say) a one-to-one function between finite equal-sized sets, hence it is also onto (and hence $g$ is actually a two-sided inverse).

We must be careful, however: sometimes the reason for constructing a bijection in the first place is in order to show that $A$ and $B$ have the same size! This kind of thing is common in combinatorics. In that case one really must show a two-sided inverse, even when $A$ and $B$ are finite; otherwise you end up assuming what you are trying to prove.

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I’ll leave you with one more to ponder. Suppose $f : A \to B$ is one-to-one, and there is another function $g : B \to A$ which is also one-to-one. We don’t assume anything in particular about the relationship between $expressvnp中文官网$ and $expressvnp中文官网$ . Are $f$ and $g$ necessarily bijections?

Posted in logic, proof | Tagged bijection, finite, function, injection, invertible, one-to-one, onto, proof, surjection | 7 Comments

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Posted on March 14, 2024 by Brent

Several commenters correctly answered the question from my previous post: if we have a function $expressvnp中文官网$ and $g : B \to A$ such that $g(f(a)) = a$ for every $a \in A$ , then $f$ is not necessarily invertible. Here are a few counterexamples:

Commenter Buddha Buck came up with probably the simplest counterexample: let $A$ be a set with a single element, and $expressvnp中文官网$ a set with two elements. It does not even matter what the elements are! There’s only one possible function $g : B \to A$ , which sends both elements of $B$ to the single element of $A$ . No matter what $f$ does on that single element $a \in A$ (there are two choices, of course), $g(f(a)) = a$ . But clearly $f$ is not a bijection.
Another counterexample is from commenter designerspaces: let $f : \mathbb{N} \to \mathbb{Z}$ be the function that includes the natural numbers in the integers—that is, it acts as the identity on all the natural numbers (i.e. nonnegative integers) and is undefined on negative integers. $g : \mathbb{Z} \to \mathbb{N}$ can be the absolute value function. Then $expressvnp中文官网$ whenever $n$ is a natural number, but $f$ is not a bijection, since it doesn’t match up the negative integers with anything.

Unlike the previous example, in this case it is actually expressvnp中文官网 to make a bijection between $expressvnp中文官网$ and $\mathbb{Z}$ , for example, the function that sends even $n$ to $n/2$ and odd $n$ to $-(n+1)/2$ .
Another simple example would be the function $f : \mathbb{N} \to \mathbb{N}$ defined by $f(n) = 2n$ . Then the function $expressvnp中文官网$ satisfies the condition, but $f$ is not a bijection, again, because it leaves out a bunch of elements.
Can you come up with an example $f : \mathbb{R} \to \mathbb{R}$ defined on the real numbers $expressvnp中文官网$ (along with a corresponding $g$ )? Bonus points if your example function is continuous.

All these examples have something in common, namely, one or more elements of the codomain that are not “hit” by $f$ . Michael Paul Goldenberg noted this phenomenon in general. And in fact we can make this intuition precise.

Theorem. If $f : A \to B$ and $g : B \to A$ such that $expressvnp中文官网$ for all $a \in A$ , then $f$ is injective (one-to-one).

expressvnp中文官网. Suppose for some $expressvnp中文官网$ we have $f(a_1) = f(a_2)$ . Then applying $g$ to both sides of this equation yields $g(f(a_1)) = g(f(a_2))$ , but because $g(f(a)) = a$ for all $a \in A$ , this in turn means that $a_1 = a_2$ . Hence $f$ is injective.

Corollary. since bijections are exactly those functions which are both injective (one-to-one) and surjective (onto), any such function $expressvnp中文官网$ which is not a bijection must not be surjective.

And what about the opposite case, when there are functions $expressvnp中文官网$ and $expressvnp中文官网$ such that $f(g(b)) = b$ for all $expressvnp中文官网$ ? As you might guess, such functions are guaranteed to be surjective—can you see why?

expressvnp中文官网 logic | Tagged bijection, function, expressvnp中文官网, invertible, one-to-one, onto, surjection | 1 Comment

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Posted on March 3, 2024 by Brent

Suppose we have sets $A$ and $B$ and a function $f : A \to B$ (that is, $f$ ’s domain is $A$ and its codomain is $expressvnp中文官网$ ). Suppose there is another function $expressvnp中文官网$ such that $expressvnp中文官网$ for every $expressvnp中文官网$ . Is $f$ necessarily a bijection? That is, does $f$ necessarily match up each element of $A$ with a unique element of $B$ and vice versa? Or put yet another way, is $f$ necessarily invertible?

If yes, prove it!
If no, provide a counterexample! For bonus points, what additional assumptions could we impose to make it true?

Posted in logic | Tagged bijection, function, intuition, invertible | 12 Comments

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Posted on February 28, 2024 by Brent

[Disclosure of Material Connection: MIT Press kindly provided me with a free review copy of this book. I was not required to write a positive review. The opinions expressed are my own.]

Beautiful Symmetry: A Coloring Book about Math
Alex Berke
The MIT Press, 2024

Alex Berke’s new book, expressvnp中文官网, is an introduction to basic concepts of group theory (which I’ve written about before) through symmetries of geometric designs. But it’s not the kind of book in which you just read definitions and theorems! First of all, it is actually a coloring book: the whole book is printed in black and white on thick matte paper, and the reader is invited to color geometric designs in various ways (more on this later). Second, it also comes with a web page of interactive animations! So the book actually comes with two different modes in which to interactively experience the concepts of group theory. This is fantastic, and exactly the kind of thing you absolutely need to really build a good intuition for groups.

The book is not, nor does it claim to be, a comprehensive introduction to group theory; it focuses exclusively on groups that arise as physical symmetries in two dimensions. It first motivates and introduces the definitions of groups and subgroups, using 2D point groups (cyclic groups $C_n$ and dihedral groups $D_n$ ) and then going on to catalogue all frieze and wallpaper groups (all the possible types of symmetry in 2D), which I very much enjoyed learning about. I had heard of them before but never really learned much about them.

One thing I really like is the way Berke characterizes subgroups by means of breaking symmetry via coloring; I had never really thought about subgroups in this way before. For example, consider a simple octagon:

An octagon has the symmetry group $expressvnp中文官网$ , meaning that it has rotational symmetry (by $expressvnp中文官网$ of a turn, or any multiple thereof) and also reflection symmetry (there are $8$ different mirrors across which we could reflect it).

However, if we color it like this, we break some of the symmetry:

expressvnp中文官网

$1/8$ turns would no longer leave the colored octagon looking the same (it would switch the blue and white triangles). We can now only do $expressvnp中文官网$ turns, and there are only $4$ mirrors, so it has $D_4$ symmetry, the same as a square. In particular, the fact that we can color something with $D_8$ symmetry in such a way that it turns into $D_4$ symmetry tells us that $D_4$ is a subgroup of $expressvnp中文官网$ . Likewise we could color it so it only has $expressvnp中文官网$ symmetry (we can rotate by $1/2$ turn, or reflect across two different mirrors; left image below) or $D_1$ symmetry (there is only a single mirror and no turns; right below). Hence $D_2$ and $expressvnp中文官网$ are also subgroups of $expressvnp中文官网$ .

Along different lines, we could color it like this, so we can still turn it by $expressvnp中文官网$ but we can no longer reflect it across any mirrors (the reflections now switch blue and white):

This symmetry group (8 rotations only) is called $C_8$ ; we have learned that $C_8$ is a subgroup of $D_8$ . Likewise we could color it in one of the ways below:

yielding the subgroups $C_4$ , $C_2$ , and $C_1$ . Note $expressvnp中文官网$ and $expressvnp中文官网$ are abstractly the same: both feature a single symmetry which is its own inverse (a mirror reflection in the case of $D_1$ and a $180^\circ$ rotation in the case of $expressvnp中文官网$ ), although geometrically they are two different kinds of symmetry. $C_1$ is also known as the “trivial group”: the colored octagon on the right has no symmetry.

Anyway, I really like this way of thinking about subgroups as “breaking” some symmetry and seeing what symmetry is left. If you like coloring, and/or you’d like to learn a bit about group theory, or read a nice presentation and explanation of all the frieze and wallpaper groups, you should definitely check it out!

Posted in books, review | Tagged expressvnp中文官网, beauty, expressvnp中文官网, expressvnp中文官网, symmetry, theory | 1 Comment

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Posted on January 30, 2024 by Brent

In my previous posts, each drawing consisted of two offset copies of the previous drawing. For example, here are the drawings for $n=3$ and $n=4$ :

You can see how the $expressvnp中文官网$ drawing contains an exact copy of the $n=3$ drawing, plus another copy with the fourth red element added to every set. The second copy is obviously offset by one unit in the vertical direction, because every set gained one element and hence moved up one row. But how far is the second copy offset horizontally? Notice that it is placed in such a way that its second row from the bottom fits snugly alongside the third row from the bottom of the original copy.

In this case we can see from counting that the second copy is offset five units to the right of the original copy. But how do we compute this number in general?

The particular pattern I used is that for even $expressvnp中文官网$ , the second copy of the $(n-1)$ -drawing goes to the right of the first copy; for odd $n$ it goes to the left. This leads to offsets like the following:

Can you see the pattern? Can you explain why we get this pattern?

Posted in expressvnp中文官网 | expressvnp中文官网 binary, bits, cube, diagram, Hasse, hypercube, lattice, subsets | 2 Comments

Post without words #31

Posted on January 23, 2024 by Brent

expressvnp中文官网 | Posted on January 23, 2024 by Brent | Tagged binary, bits, cube, diagram, Hasse, hypercube, lattice, subsets | 2 Comments

A few words about PWW #30

Posted on January 22, 2024 by Brent

A few things about the images in my previous post that you may or may not have noticed:

As several commenters figured out, the $expressvnp中文官网$ th diagram (starting with $n = 1$ ) is showing every possible subset a set of $expressvnp中文官网$ items. Two subsets are connected by an edge when they differ by exactly one element.
All subsets with the same number of elements are aligned horizontally.
Each diagram is made of two copies of the previous diagram—one verbatim, and one with a new extra element added to every subset, with edges connecting corresponding subsets in the two copies. Do you see why this makes sense? (Hint: if we want to list all subsets of a set, we can pick a particular element and break them into two groups, one consisting of subsets which contain that element and one consisting of subsets which don’t.)
As commenter Denis pointed out, each diagram is a hypercube: the first one is a line (a 1-dimensional “cube”), the second is a square, the third is a cube, then a 4D hypercube, and so on. (On my own computer I rendered them up to $expressvnp中文官网$ but it gets very hard to see what’s going on after $5$ .)
Each subset can also be seen as corresponding to a bitstring specifying which elements are in the set. A dot corresponds to a 1, and an empty slot to a 0. So another way to think of this is the graph of all bitstrings of length $expressvnp中文官网$ , where two bitstrings are connected by an edge if they differ in exactly one bit.
Thinking of it as bitstrings makes it clearer why we get hypercubes: each bit corresponds to a dimension. So for example for the 3D case you could think of the three bits as corresponding to back/front, left/right, and down/up.
I drew something similar to this many years ago, in Post without words #2. The big difference is that it recently occurred to me how to lay out the nodes recursively to highlight the hypercube structure, so they don’t all just smoosh together on each line.
There was actually some interesting math involved in figuring out the horizontal offsets to use for the subset nodes; perhaps I’ll write about that in another post!

Posted in expressvnp中文官网 | Tagged binary, bits, cube, expressvnp中文官网, Hasse, hypercube, lattice, subsets | expressvnp中文官网

expressvnp中文官网

Posted on January 20, 2024 by Brent

Image | Posted on January 20, 2024 by expressvnp中文官网 | Tagged expressvnp中文官网, bits, cube, diagram, Hasse, hypercube, expressvnp中文官网, subsets | 5 Comments

The First Six Books of the Elements of Euclid, by Oliver Byrne (Taschen)

expressvnp中文官网 January 11, 2024 by Brent

Recently for my birthday I received a copy of Oliver Byrne’s 1847 edition of Euclid’s Elements (pictured at right), republished by Taschen Books in 2010. I’ve only just started reading it, but it’s beautiful and fascinating. expressvnp中文官网 was a 19th-century civil engineer and mathematician, best known nowadays for this incredible “color-coded” edition of Euclid. Euclid’s Elements, of course, is the most successful and influential mathematics textbook of all time, widely used as a geometry textbook even up into early 1900’s. Nowadays hardly anyone reads the Elements itself, but its content and style is still widely emulated. In 1847, Oliver Byrne decided to make an English edition of the Elements that not only used colored illustrations, but actually used color-coded pictures of lines, angles, and so on, expressvnp中文官网 to refer to the picture instead of using the traditional points labelled by letters (see the example below). I can’t imagine how much work went into designing and printing this in the mid-1800s. I guess there would have to be four engraved plates for every single page? In any case, it’s beautiful, creative, and surprisingly effective. I spent a while last night going through some of the propositions and their proofs with my 8-year-old son—I highly doubt he would have been interested or able to follow a traditional edition that used letters to refer to labelled points in a diagram.

It’s also surprisingly inexpensive—only $20! You can get a copy through Taschen’s website here.

In a similar vein, the publisher Kronecker Wallis decided to finish what Byrne started, creating a beautifully designed, artistic version of all 13 books of Euclid. (Byrne only did the first six books; I am actually not sure whether because that’s all he intended to do, or because that’s all he got around to.) Someday I would love to own a copy, but it costs 200€ (!) so I think I’m going to wait a bit…

Posted in expressvnp中文官网, geometry, pictures | Tagged beauty, Byrne, color, design, elements, Euclid, illustration | 3 Comments

The Math Less Traveled

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Post without words #31

A few words about PWW #30

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The First Six Books of the Elements of Euclid, by Oliver Byrne (Taschen)

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