How to execute a selected statement?

Logical conclusions: If no As are selected, then both Ps must be selected.?

• Now, I read somewhere that just by nature of the above, you can also say, If both Ps are not selected, then at least 1 A must be selected. The problem I'm having is figuring out how the above logically follows from the initial statement. I know it's true, based on the answer, but I just don't get the logic behind it.

• Answer:

  The initial statement tells us a number of things. Because of the use of the word "both" in relationship to the selection of P's we know that there are two P's to select. We also know that if none of the A's are selected then all the P's must have been selected. if even one of the P's was not selected then it must be that the condition "no A's are selected" must be false. If it were not false then both the P's would have been selected. When the condition "no A's are selected" is false the condition "at least one A has been selected" must be true. It therefore follows that if even one of the P's was not selected then at least one A has been selected.