Let A mean "Equation A has a solution" and B mean "Equation B has no solution." I am a little confused, so I wrote down some possibilities and I wish to see if my interpretation of the following is correct:

  1. Equations A and B both have no solutions. In symbols: ~A /\ ~B.
  2. Neither equation A nor B has a solution. In symbols: ~(A \/ B).
  3. Both equations A and B have no solutions. In symbols: ~(A \/ B).

If I'm correct, I want to know intuitively the difference between (1) and (3). Thanks.

  • The three statements are all logically equivalent, there is no difference truth-functionally. But if B is supposed to mean "Equation B has no solution", then you need to invert the negations on it. – lemontree Aug 7 '20 at 10:57

For statements 1 and 3 you are looking at examples of De Morgan's Laws.

From https://en.wikipedia.org/wiki/De_Morgan%27s_laws

The complement of the union of two sets is the same as the intersection of their complements; and The complement of the intersection of two sets is the same as the union of their complements. or

not (A or B) = not A and not B; and
not (A and B) = not A or not B

And LemontTree has a good point as well.

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