Adriano

Hey, quick question. Consider a bijection $f: A \to B$ whose inverse is $g: B \to A$, and suppose that $C \subseteq A$. Is it true that the preimage $g^{-1}(C)$ is equivalent to the image $f(C)$? I don't want to confuse the notation for inverses ($f = g^{-1}$) with the notation for preimages of sets ($g^{-1}(C)$). Basically, I'm wondering if $(g^{-1})^{-1}(C) = C$.

Happy new year! =]

Hey, quick question. Suppose that $A \subseteq B$ and $|A|=n=|B|$ for some finite $n \in \mathbb N$. Then why must $A=B$? Should I be using some kind of pigeonhole argument?

@PedroTamaroff It seems to work out. =]

@PeterTamaroff Ooh, thanks. =]

@PeterTamaroff I guess it doesn't have to be unique. Although I had been thinking about the problem of constructing a poset that contains a unique maximal element but no greatest element.

Hey, quick question. Given any non-empty finite set $S$ and partial order relation $R$ on $S$, must $S$ always contain a maximal element? Is it the case that the only way for for $S$ to not have a maximal element is if $S$ is infinite (for example, when $S=\Bbb{Z}$ and $R=~>$)?

We have: \begin{align*} (x - 2)(x - 3)(x + 5) &= (x^2 - 5x + 6)(x + 5)\\ &= (x^3 - 5x^2 + 6x) + (5x^2 - 25x + 30)\\ &= x^3 - 19x + 30 \end{align*}

Note that you should get a $-19x$ term (and not a $-9x$ term) because: $$ 6x - 10x - 15x = -19x $$ Once you have that, just divide both sides of the equation by $2$ and subtract the $60$ to the other side.

Expand and bring everything to one side to get: $$ 0 = x^3 - 19x - 30 $$ then guess some factors (use the rational root theorem or factor theorem), then perform long division to factor completely.

Yes, you made an error. Did you get $a = 2$? As a hint, note that $f(5) = 120$. There are two other values of $x$ such that $f(x) = 120$.

Substitute $x = 4$ and $y = 36$ in order to solve for $a$. Once you know what $a$ is, you can then substitute $y = 120$ in order to solve for $x$.

A semicircle is a curve. The base of the triangle is a line. When a line intersects a curve, it makes a point.

Which part?

Similar triangles is more complicated. If we use similar triangles, what are the dimensions of your two triangles?

So does 2y make sense yet?

The triangles have different sizes.

We are only looking at the base of the 3D solid. No heights yet.

So y represents the vertical distance from point B to the x-axis.

I am defining the point B(x,y) to be the point where the triangle intersects the upper semicircle.

Remind me what y represents.

It's a 3D shape. It's drawn from a perspective.

Why doesn't it make sense?

I don't understand your question. Can you elaborate?

See the picture on the right? A and B aren't on the y-axis.

A and B are both on the xy-plane, but they don't have to be in the center.

y is the vertical distance from the x-axis to the point B.

If y = 1, then the base is 2(1) = 2.

It's 2y.

No.

Does it make sense that the length of the base is 2y?

So we just add them together. y + y = 2y.

Or can you see that the distance from B to the x-axis is y?

Can you see that the distance from A to the x-axis is y?

Can you see that the length must be 2y?

Can you see that the points are A(x,-y) and B(x,y)?

Can you see points A and B on the diagram?

Can you see that the points are A(x,-y) and B(x,y)?

Can you see that the length must be 2y?

We want to find the length of the base (the length of AB).

Here's how I would do it.

Hmm that's not quite right.

Can you explain what the (2-y) represents?

It's a straight pink line, AB.

We're only looking at the base of the 2D triangle.

I'm focusing on the points A(x,-y) and B(x,y).