"$ab=0$ implies that $a=0$ or $b=0$", does this mean that we can have $a=b=0$? I dont understand if or means that if $a=0$ then $b \not= 0$?
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In math, "or" always means inclusive or. In this case, "$a = b = 0$" is valid for "$a = 0$ or $b = 0$".
If a is $0$ or b is $0$ or they are both $0$, the statement is true.
$a=0$ then $b \not= 0$ This is and exclusive or then the table would look like this: