If and only if relationship between them?

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To show that two definitions are equivalent, it is necessary to establish an "if and only if" relationship between them. This means proving that each definition implies the other, which is crucial for demonstrating their equivalence. The discussion highlights the distinction between equivalence and identity in mathematics, where equivalence involves variables that can yield identity under certain conditions. The conversation emphasizes that proving this logical relationship is essential when addressing problems of equivalence. Thus, confirming an "if and only if" relationship is key to solving such mathematical problems.
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"if and only if" relationship between them?

Hi Pfers, just a little question :

If you have a problem of the form "Show that these definitions are equivalent",do I basically have to prove an "if and only if" relationship between them?
 
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Equivalence is different from identity in math, in that, if two sides of an equation are equivalent, as opposed to identical, it just means that there are variables involved of which there are only certain values (in place of the variable(s)) that make the equation a case of identity. So, 3+3 = 6 is a case of identity, while 3 + x = 6 is a case of equivalence, since there is a variable involved (to put it simply). x * 0 = 6 is a case of non-equivalence, since both sides of the equation can never be identical.

You seem to be talking about logic, though (and answering a question), so, yes, you'll want to show that an "if and only if" is possible.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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