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I just had my first class of calculus where we mostly talked about sets. We defined the subset like this:

Let A and B be sets. We say that A is a subset of B if ##x \in A \implies x \in B##

But if we look at the truth table of the implication (https://en.wikipedia.org/wiki/Truth_table#Logical_implication) we see that if the first statement if false, then the implication is true independently of the second statement. So if A and B are disjoint sets and as ##x## is a free variable we can choose it in a way that it is not in A, then the implication will be true, but this clearly contradicts with the concept of a subet relation. Don't we need a universal quantifier in this definition?