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glb_lub

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1) I am having trouble understanding some parts of congruence as is explained by T Apostol in the book Calculus volume 1.

In the chapter on integration , while discussing area functions he explains congruence as follows :-

Now I can appreciate that concepts such as 'area' , 'congruence' are being defined in terms of sets. But , the definition of congruence makes use of the notion of 'distance' . And while calculating distance between two points one has to make use of the Pythagoras theorem. And as far as I know any proof of Pythagoras theorem makes use of congruence of triangles in one way or the other. I don't know whether I am being fussy here but aren't we having circularity of reasoning here. Or are there proofs of Pythagoras theorem not depending on congruence of triangles (or for that matter notions of distance not making use of Pythagoras theorem).

2) Furthermore , he defines rectangles as those sets of points which are congruent to a set of the form

S = {(x,y)| 0≤x≤h , 0≤y≤k}

where h and k are lengths of the edges of the rectangle.

Later by an axiom , area of a rectangle is taken as hk.

My question is , is the area 'hk' according to the axiom* only for the set S ? If so,do we have to show that more general rectangles are 'congruent' to the set S , by making use of 'congruence' as defined in terms of sets. (by general rectangles I mean rectangles which are rotated and/or translated versions of the rectangle represented by set S).

And do we have to do this also for other geometric objects such as triangles?* axiom 5 - choice of scale - Every rectangle is measurable . If a rectangle has lengths of edges h and k then its area is hk.Overall , I am having some trouble in seeing the motivation behind defining congruence and rectangles as is done in the book.

In the chapter on integration , while discussing area functions he explains congruence as follows :-

Congruence is used here in the same sense as in elementary Euclidean geometry. Two sets are said to be congruent if their points can be put in one-to-one correspondence in such a way that distances are preserved. I.e ,if 2 points p and q in one set correspond to p' and q' in the other, the distances from p to q will be equal to the distance between p' and q';this must be true for all choices of p and q.

Now I can appreciate that concepts such as 'area' , 'congruence' are being defined in terms of sets. But , the definition of congruence makes use of the notion of 'distance' . And while calculating distance between two points one has to make use of the Pythagoras theorem. And as far as I know any proof of Pythagoras theorem makes use of congruence of triangles in one way or the other. I don't know whether I am being fussy here but aren't we having circularity of reasoning here. Or are there proofs of Pythagoras theorem not depending on congruence of triangles (or for that matter notions of distance not making use of Pythagoras theorem).

2) Furthermore , he defines rectangles as those sets of points which are congruent to a set of the form

S = {(x,y)| 0≤x≤h , 0≤y≤k}

where h and k are lengths of the edges of the rectangle.

Later by an axiom , area of a rectangle is taken as hk.

My question is , is the area 'hk' according to the axiom* only for the set S ? If so,do we have to show that more general rectangles are 'congruent' to the set S , by making use of 'congruence' as defined in terms of sets. (by general rectangles I mean rectangles which are rotated and/or translated versions of the rectangle represented by set S).

And do we have to do this also for other geometric objects such as triangles?* axiom 5 - choice of scale - Every rectangle is measurable . If a rectangle has lengths of edges h and k then its area is hk.Overall , I am having some trouble in seeing the motivation behind defining congruence and rectangles as is done in the book.

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