Questions on equivalent sets and Cartesian products

Bob_McMillan

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Aug 12, 2017
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First question, what are equivalent sets? My prof said that sets are equivalent if they have the same cardinality, but a few other sources I have read say that equivalency means having the exact same elements in both sets.

Second, can two non-equivalent sets (going my my prof's definition) have a Cartesian product?

The problem (simplified to the final steps) is A x B, where A = {a,e,g.n} and B = {s,q,u,a,r,e}. Obviously n(A)=4 and n(B)=6, so I'm guessing that there is no Cartesian product and the answer is null?
 
Well, it's true that equivalent sets have will always have the same cardinality, but that's not why they're equivalent. It's sort of like how planes are made of aluminum, but that's not what makes them airplanes. Aluminum foil is also made of aluminum but it is clearly not an airplane ;). The other sources that say that two sets are equivalent if they have all of the same elements is the correct definition. More specifically, the rigorous definition of equivalent sets is:

The notation A = B is shorthand to state that the following two formulae hold:

\(\displaystyle \forall X \left[ \left(X \in A \right) \iff \left(X \in B \right) \right]\)

\(\displaystyle \forall Y \left[ \left(A \in Y \right) \iff \left(B \in Y \right) \right]\)

Essentially, for two sets to be equivalent, any element (X) of one set must also be an element of the other, and if one set is a subset of some third set (Y), the other set must be too. As for the Cartesian product, the definition of that is:

For all sets A and B, the Cartesian Product of A and B is A x B (read "A cross B"), consisting of all ordered pairs (X, Y) with \(\displaystyle X \in A\) and \(\displaystyle Y \in B\)

This, then, doesn't require the sets to be equivalent, nor even have the same cardinality. As an example, let \(\displaystyle A = \left\{ 1, 3, 5 \right\}\) and \(\displaystyle B = \left\{ 2, 4 \right\}\). Given these, \(\displaystyle A \times B = \left\{ (1, 2), (1, 4), (3, 2), (3, 4), (5, 2), (5,4) \right\}\)

Hopefully this helps you understand the concepts a bit better.
 
Building on what the prior post said, I suspect that what your teacher meant is

"two equivalent sets have the same cardinality."

That statement is correct. It does not imply that two sets with the same cardinality are equivalent.

All mothers are women, but not all women are mothers.
 
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