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Two sets are equivalent if they have the same cardinality.
For finite sets it simply means that the two sets have the same number of distinct elements.
Thus {1, 2, 3, 3} is equivalent to {a, b, b, b, b, c, c}; each set has cardinality 3.
For infinite sets, it is a bit more complicated. If you can define a bijective mapping (one-to-one and onto) from the elements of one set to the elements of the other then the two sets are equivalent. By definition, bijective mappings must have an inverse so defining the mapping in one direction is sufficient.
It is relatively easy to see that the set of odd integers is equivalent to the set of even integers if you consider the mapping f(x) = x+1 where x is odd. Or perhaps f(x) = x + 7
What is less obvious is that the set of all integers is equivalent to the set of even integers when you consider the mapping f(x) = 2x where x is any integer.
Thus it is possible for a proper subset of an infinite set to be equivalent to the superset!
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