# Is my interpretation of “and” correct in these statements?

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.

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.