Equivalence relation proof and notation

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December 3rd 2012, 08:07 AM
carla1985

Equivalence relation proof and notation

I missed a lecture last week and am now struggling with my homework. This is the first question (and the easiest lol):

1. Denote by D the set of all triangles in the plane.
Define for ABC; A'B'C' in D the relation ABC ~_s A'B'C' if and only if ABC is similar to A'B'C'.

Prove that ~_s is an equivalence relation on D.

I've got my head around the basics of equivalence relations, I think but firstly I dont understand the notation my lecturer has used (~_sand ~_c ~₂ in other questions). If someone could explain the notation and give me a little help with this question I would be very grateful, and then I could do the rest of them myself :)

Thank you
December 3rd 2012, 08:28 AM
Plato

Re: Equivalence relation proof and notation

Quote:

Originally Posted by carla1985

1. Denote by D the set of all triangles in the plane.
Define for ABC; A'B'C' in D the relation ABC ~_s A'B'C' if and only if ABC is similar to A'B'C'.
Prove that ~_s is an equivalence relation on D.

It is impossible to know why one notation is used over another.

In most geometry courses $\Delta ABC \approx \Delta PQR$ is used for similar triangles. It seems that you instructor uses $\Delta ABC \sim _S \Delta PQR$ in its place.

In general, relations are sets of ordered pairs.
We write $(x,y)\in\mathcal{R}$ to say that $x\text{ is }\mathcal{R}\text{ related to }y$ .

Now that is often written as $x\mathcal{R}y$ . But it appears that your instructor would prefer to use $x\sim_{\mathcal{R}}y~.$
December 3rd 2012, 09:39 AM
carla1985

Re: Equivalence relation proof and notation

Ah I understand, in the lecture notes he used xRy, which I understand, it threw me when he changed notation. Thanks for the help x
December 3rd 2012, 11:48 AM
Plato

Re: Equivalence relation proof and notation

Quote:

Originally Posted by carla1985

Ah I understand, in the lecture notes he used xRy, which I understand, it threw me when he changed notation. Thanks for the help x

Is it true that any triangle is similar to itself?

If $\Delta~I\sim_S\Delta~II$ is true that $\Delta~II\sim_S\Delta~I~?$

If $\Delta~I\sim_S\Delta~II~\&~\Delta~II\sim_S\Delta~I II$ is true that $\Delta~I\sim_S\Delta~III~?$
December 3rd 2012, 12:04 PM
carla1985

Re: Equivalence relation proof and notation

I think I've got it. My answer is:

For all ABC, A'B'C' and A''B''C'' in D

Reflexive: A triangle is similar to itself since AB/AB=BC/BC=AC/AC
so ABC ~s ABC

Symmetric: If ABC and A'B'C' are similar then its corresponding angles are equal so
if ABC ~s A'B'C' then A'B'C' ~s ABC

Transistive: If ABC is similar to A'B'C' then their corresponding angles are equal, and if A'B'C' is similar to A''B''C'' then its corresponding angles are are also equal.Therefore the corresponding angles of ABC and A''B''C'' must be equal so they must also be similar.
Hence if ABC ~s A'B'C' and A'B'C' ~s A''B''C'' then ABC ~s A''B''C''

So ~s is an equivalence relation on D.

I'm assuming thats enough for the proof? Thanks