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difference between congruence and equality

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what is the difference between the following statements:

 

a congruent (triple equal sign) to b mod (n)

and

a=b mod(n)??

 

do they differ in any properties ?

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what is the difference between the following statements:

 

a congruent (triple equal sign) to b mod (n)

and

a=b mod(n)??

 

do they differ in any properties ?

 

 

Those two statements are synonymous.

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I've taken from the literature that it is most appropriate to say that equivalent objects are the same objects. As such equality is a subset of the class of all equivalence relations(congruent relations.) I guess what I am getting at is, wouldn't it be a bit inappropriate to assume that two objects that are congruent are in fact equal? And so although they may be synonymous would it not be problematic if one were to treat them as having the same meaning?

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I've taken from the literature that it is most appropriate to say that equivalent objects are the same objects. As such equality is a subset of the class of all equivalence relations(congruent relations.) I guess what I am getting at is, wouldn't it be a bit inappropriate to assume that two objects that are congruent are in fact equal? And so although they may be synonymous would it not be problematic if one were to treat them as having the same meaning?

 

What is synonymous are the statements "a is congruent to b mod n" and "a=b (mod n)".

 

Congruence is not equality.

 

However, when one is working with equivalence classes, one can correctly state that two equivalence classes are equal. It is often the case in mathematics that one is working with equivalence classes, and equality is equality of equivalence classes.

 

Sometimes the fact that it is equivalence classes that are the objects of interest is understood but not explicitly stated -- as, for, instance in the theory of measure and integration where "equals" often means "equals almost everywhere" -- equivalence meaning equality except possibly on a set of measure zero.

 

Whether the use of equivalence classes is explicitly stated or just understood depends on the intended audience. It is made explicit in introductory tests. It is often implicit when one is addressing experts.

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