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Math Help - What suffices to prove two properties equivalent?

  1. #1
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    What suffices to prove two properties equivalent?

    Hello. I took several math courses in college and did some simple proofs, but now I'm trying to re-learn the material with Apostol's Calculus. In some of his problems (i.e. 1.15.8, 1.15.9), he asks that you show that properties (here, equalities) of integrals are 'equivalent,' or 'may be expressed in equivalent form.' Does this entail anything more than applying the one of the said properties to the left side of the other to obtain the right side of the other, and vice versa?

    For example, if I am given the property x(a + b) = a/(1/x) + b/(1/x) and asked to show that it is equivalent to the property (a + b)/x = a(1/x) + b(1/x), is it sufficient to write something like this:

    x(a + b) [left side of the first property]
    = (a + b) / (1/x) [preparing to apply the second property]
    = a(1/(1/x)) + b(1/(1/x)) [applying the second property]
    = ax + bx [simplifying]
    = a/(1/x) + b/(1/x) [right side of the first property]

    I realize that the above example is very contrived, but I hope you get the gist of what I'm asking. Essentially: what suffices to prove the equivalence of two properties?
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  2. #2
    Senior Member Tinyboss's Avatar
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    Don't break up each property into a left and right side. Rather, accept the first property as true, then use it to prove the second--in your example, take the first equation and divide by x^2 to get the second one. Then start over, accepting only the second property as true, and use it to prove the first (in this case, by multiplying by x^2).

    In other words, two propositions are equivalent if each implies the other.
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