Ahh! I see. I don't know why I was thinking you meant nand !(P and Q). I misunderstood your notation I guess.

The student's teacher would say ...

1 + 1 = 2 and 2 + 2 = 4 and 2 * 2 = 4 but 1 * 1 != 2

... which could be written as ...

1 + 1 = 2 and 2 + 2 = 4 and 2 * 2 = 4 and not (1 * 1 = 2)

I get it now.