L1.8: Relations

[Guganesan S. Ilavarasan]

Good morning everyone,

           While revising I found this long time forgotten dobut to be asked in 'Relations' lecture. Why does Prof. Mukund state that a Square's corners as a subset of R2 x R2 x R2 x R2? Is he saying like the point as (AxB)? as in (AxB) X (BxC) X (CxD) X (DxA)? But how could he simply square an ambiguous AxB?

Thanks and best wishes.

[Anand Iyer]

R x R represents each pair.  R2 is same as R x R.

Now, there are 4 such pairs, each for each corner.  Now, consider one set consisting of these 4 pairs.  This is obviously a subset of R2 x R2 x R2 x R2.

Essentially, symbol x represents cartesian-product, not multiplication.

[Guganesan S. Ilavarasan]

Thank you for the reply, Mr Anand, so it's like for eg, one of R X R / R2 is (2,2)? so R2 X R2 X R2 X R2 = (0X0) X (0X2) X (2X2) X (2X0) ? or have I misinterpreted again?

[Anand Iyer]

not really.

We're only saying that the 4 pairs of the corners (0, 0), (0, 2), (2,2) and (2,0) make a set, and that set is a subset of the R2 X R2 X R2 X R2.

So, think of it like this.  R2 is the (cartesian) set comprising of all pairs of real numbers.  R2 X R2 X R2 X R2 is a (cartesian) set that comprises 4 pairs.  So, our set of corners are a subset of this.  That's all we're saying by this.

To give an example, let's forget that R is a set of reals and for our convenience say R = {0,1,2}

R2 = {(0,0), (0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)}.  Note that there are 9 elements in this.

Now, R2 X R2 X R2 X R2 will have 9 * 9 = 81 elements in it.  I'll not even start to write, because there're so many.

Obviously, now, the 4 pairs of squares are part of the 81-element set.

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Cheers,

Anand

[Mathematics 1 Support 2]

Nice explanation Anand.

[Guganesan S. Ilavarasan]

Good morning Mr Anand,

                          That explanation was just phenomenal friend; a true eye-opener indeed. The way Prof Mukund sir took a R X R into an R^2 confused me a little, but since both are infinity don't think that makes much difference. Sorry to trouble you with yet another doubt friend, in the previous example explaining Phytagarous theorem with relations, sir took that x,y,z  c N X N X N. Why didn't he take the same as a subset of R^2 X R^2 X R^2 brother? Since each point is not a pair of points x,y like the square since they are not represented on a plane/graph as the square is represented in the lecture for the example, Mr Anand?

Heartiest regards.

[Guganesan S. Ilavarasan]

And sorry to ask further sir, just a small doubt while comprehension the example, if R = (0,1,2) then R^2 is 9 as you told sir. Then R^2 X R^2 is 81, again as you told sir. Then R^2 X R^2 X R^2 is which is 9 X 9 X 9 X 9, 81 X 81 so the total elements are 6,561 isn't it sir? Sorry not to mean to ask but thought of clearing the concern, sir.

Thanks in advance.

[Anand Iyer]

yes, Guganesan!  and, thanks @Aditya also pointing out the mistake. 

R2 * R2 * R2 * R2 will have 81 * 81 elements in it.  

[Anand Iyer]

As for the other question about the Pythagoras theorem, can you tell me which lecture you're referring to?

[Anand Iyer]

Ok, I think you're referring to the same slide where Pythagorean triples were mentioned.  

This is only about one triplet.  One triplet is formed from 3 integers - N x N x N.  It's unlike the case with square, where there are 4 corners.  Hence, they're treated differently...

Hope that was understandable...